Integrated Math I Honors

Unit 1 25 topics

Equations & Inequalities

Solving for an unknown, modeling with it, and describing ranges of answers.

1.1 Linear Equations

1.1.1
Solving Linear Equations With Unknown Coefficients
Treat the other letters as numbers and isolate the variable you want. Divide by its coefficient only when that coefficient can’t be zero.
e.g. Solve $ax - 3 = 7$ for $x$ : $x = \tfrac{10}{a}$
$\displaystyle ax + b = c$
$\displaystyle \Rightarrow\; x = \frac{c - b}{a} \quad (a \ne 0)$
1.1.2
Solving Linear Equations With Unknown Coefficients by Factoring
When the variable appears in more than one term, gather those terms, factor the variable out, then divide.
e.g. $kx + 2x = 9 \Rightarrow x = \tfrac{9}{k + 2}$
$\displaystyle ax + bx = c$
$\displaystyle \Rightarrow\; x(a + b) = c$
$\displaystyle \Rightarrow\; x = \frac{c}{a + b}$
1.1.3
Solving Linear Inequalities With Unknown Parameters
Same steps as an equation. If you divide by a parameter, its sign decides whether the inequality flips.
$\displaystyle ax > b \;\Rightarrow\; \begin{cases} x > \frac{b}{a} & a > 0 \\[2pt] x < \frac{b}{a} & a < 0 \end{cases}$
1.1.4
Solving Many-Variable Equations
To solve a formula for one variable, undo the operations around it until it stands alone.
e.g. $P = 2l + 2w \Rightarrow w = \tfrac{P - 2l}{2}$
$\displaystyle A = \tfrac{1}{2}bh \;\Rightarrow\; h = \frac{2A}{b}$

1.2 Modeling With Linear Equations

1.2.1
Interpreting Linear Expressions
In $mx + b$ , $b$ is the fixed starting amount and $m$ is the amount added for each unit of $x$ .
e.g. $25 + 0.15m$ : a $25 fee plus 15¢ per mile
$\displaystyle \text{total} = \underbrace{m}_{\text{per unit}}\,x + \underbrace{b}_{\text{fixed}}$
1.2.2
Modeling With Linear Equations
Name the unknown, write the equation the words describe, solve, and check that the answer makes sense.
e.g. $30 to join plus $10 a month, $110 spent → $30 + 10m = 110$ , $m = 8$ months
1.2.3
Consecutive Integer Problems
Call the first integer $n$ . Consecutive integers are $n, n+1, n+2$ . Consecutive even or odd integers are $n, n+2, n+4$ .
e.g. Three consecutive integers sum to 48 → $3n + 3 = 48$ , $n = 15$
$\displaystyle n + (n + 1) + (n + 2) = 3n + 3$
1.2.4
Speed, Distance, Time Problems
Write $d = st$ for each traveler, then connect them: same distance, same time, or distances that add up.
e.g. Cars leave together at 50 and 60 mph in opposite directions → $110t = 330$ when 330 mi apart, $t = 3$ h
$\displaystyle d = s\,t$
1.2.5
Further Speed, Distance, Time Problems
A chaser closes a gap at the difference of the speeds. Travelers heading toward each other close it at the sum.
$\displaystyle t = \frac{\text{gap}}{s_1 - s_2} \;\text{(catching up)}$
$\displaystyle t = \frac{\text{gap}}{s_1 + s_2} \;\text{(approaching)}$
1.2.6
Modeling Work Problems
Each worker does a fraction of the job per hour. Working together, the rates add.
e.g. Pipes fill a tank in 3 h and 6 h → $\tfrac13 + \tfrac16 = \tfrac12$ → together 2 h
$\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$
1.2.7
Modeling Mixture Problems
Track the pure ingredient: concentration times volume, added across the parts.
e.g. 10 L at 20% plus $x$ L at 50% makes 30% → $2 + 0.5x = 0.3(10 + x)$ , $x = 5$
$\displaystyle c_1V_1 + c_2V_2 = c\,(V_1 + V_2)$

1.3 Linear Inequalities

Closed dot and square bracket: the endpoint is included. Open dot and parenthesis: it is not. Infinity always takes a parenthesis.

1.3.1
Solving Compound Inequalities
For $a < bx + c < d$ , do each operation to all three parts. Dividing by a negative flips both signs.
$\displaystyle -1 < 2x + 3 < 9$
$\displaystyle \Rightarrow\; -4 < 2x < 6$
$\displaystyle \Rightarrow\; -2 < x < 3$
1.3.2
Interval Notation
A square bracket includes the endpoint; a parenthesis excludes it.
$\displaystyle [a, b]:\ a \le x \le b$
$\displaystyle (a, b):\ a < x < b$
$\displaystyle [a, b):\ a \le x < b$
1.3.3
Unbounded Intervals
Infinity always takes a parenthesis, because it is not a number you can reach.
$\displaystyle x \ge 2 \;\to\; [2, \infty)$
$\displaystyle x < 5 \;\to\; (-\infty, 5)$
$\displaystyle \text{all reals} \;\to\; (-\infty, \infty)$
1.3.4
Unions of Intervals
$\cup$ means “or”: everything in either interval.
$\displaystyle (-\infty, 1) \cup [4, \infty)$
1.3.5
Intersections of Intervals
$\cap$ means “and”: only the part both intervals share.
$\displaystyle [0, 5] \cap (3, 8] = (3, 5]$
1.3.6
Compound OR Inequalities
Solve each part on its own. The answer is the union.
$\displaystyle x < -2 \;\text{or}\; x \ge 1$
$\displaystyle \to\; (-\infty, -2) \cup [1, \infty)$
1.3.7
Compound AND Inequalities
Solve each part. The answer is the overlap, and if nothing overlaps there is no solution.
$\displaystyle x > -1 \;\text{and}\; x \le 4 \;\to\; (-1, 4]$
1.3.8
Introduction to Modeling With Inequalities
Translate the words: at least ≥, at most ≤, no more than ≤, more than >, fewer than <.
e.g. “At least 18 years old” → $a \ge 18$
1.3.9
Modeling With One-Step Inequalities
Write it, solve it, then round in the direction the situation demands.
e.g. Bags hold 12 lb, 50 lb of sand → $12b \ge 50$ , $b \ge 4.17$ → 5 bags
1.3.10
Modeling With Two-Step Inequalities
A fixed amount plus a rate, compared with a limit.
e.g. $40 budget, $10 entry plus $6 per ride → $10 + 6r \le 40$ , $r \le 5$
$\displaystyle b + rx \le L$
1.3.11
Modeling With Compound Inequalities
A quantity that must stay inside a range gives a three-part inequality.
e.g. Keep milk between 35°F and 40°F → $35 \le T \le 40$

1.4 Nonlinear Equations

1.4.1
Solving Equations Using the Square Root Method
Isolate the square, then take the square root of both sides. Keep the negative root.
e.g. $(x - 3)^2 = 16 \Rightarrow x - 3 = \pm 4 \Rightarrow x = 7$ or $-1$
$\displaystyle x^2 = k \;\Rightarrow\; x = \pm\sqrt{k} \quad (k \ge 0)$
1.4.2
Solving Equations With Odd Exponents Using the Nth Root Method
An odd power has exactly one real root, and it can be negative.
e.g. $x^3 = -64 \Rightarrow x = -4$
$\displaystyle x^n = k,\ n \text{ odd} \;\Rightarrow\; x = \sqrt[n]{k}$
1.4.3
Solving Equations With Even Exponents Using the Nth Root Method
An even power has two real roots when $k > 0$ , one when $k = 0$ , and none when $k < 0$ .
e.g. $x^4 = 81 \Rightarrow x = \pm 3$
$\displaystyle x^n = k,\ n \text{ even},\ k > 0$
$\displaystyle \Rightarrow\; x = \pm\sqrt[n]{k}$

Unit 2 32 topics

Two-Variable Equations & Inequalities

Lines, the situations they model, and where they cross.

2.5 Graphs of Linear Equations

Pick the form by what you know: slope and intercept, slope and a point, or a quick way to both intercepts.

2.5.1
Equations of Lines in Slope-Intercept Form
$m$ is the slope and $b$ is the $y$ -intercept.
e.g. $y = -2x + 5$ : slope $-2$ , crosses the $y$ -axis at $(0, 5)$
$\displaystyle y = mx + b$
$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
2.5.2
Properties of Lines Given in Slope-Intercept Form
Positive $m$ rises, negative $m$ falls, zero is flat. Larger $\lvert m \rvert$ means steeper. The $x$ -intercept is where $y = 0$ .
$\displaystyle x\text{-intercept} = -\frac{b}{m}$
2.5.3
Equations of Lines in Point-Slope Form
Use it when you know the slope and one point on the line.
e.g. Slope 3 through $(2, -1)$ : $y + 1 = 3(x - 2)$
$\displaystyle y - y_1 = m(x - x_1)$
2.5.4
Equations of Lines in Standard Form
Integer coefficients with $x$ and $y$ on the same side. The quickest form for finding both intercepts.
e.g. $2x + 3y = 12$ : intercepts $(6, 0)$ and $(0, 4)$
$\displaystyle Ax + By = C$
2.5.5
Properties of Lines Given in Standard Form
Read the slope and intercepts straight from the coefficients.
$\displaystyle m = -\frac{A}{B}$
$\displaystyle x\text{-int} = \frac{C}{A} \qquad y\text{-int} = \frac{C}{B}$

2.6 Modeling With Two-Variable Linear Equations

On a distance–time graph the slope is the speed. On a speed–time graph the slope is the acceleration, and the shaded area is the distance traveled.

2.6.1
Modeling With Linear Equations in Two Variables
Find the starting value (the intercept) and the rate (the slope), then write $y = mx + b$ .
e.g. Plant 4 cm tall growing 1.5 cm a week: $h = 1.5w + 4$
2.6.2
Modeling Totals Using Two-Variable Linear Equations
Two kinds of item with a fixed total: price times count for each, added.
e.g. Adult tickets $12, child $8, $200 total: $12a + 8c = 200$
$\displaystyle ax + by = T$
2.6.3
Analyzing and Interpreting Graphs of Linear Equations
The slope is the rate of change, in $y$ -units per $x$ -unit. The $y$ -intercept is the value when $x = 0$ . The $x$ -intercept is where the quantity hits zero.
2.6.4
Further Modeling With Linear Equations in Two Variables
From two data points, compute the slope, then use point-slope form.
e.g. $50 for 2 h, $80 for 4 h → $m = 15$ , $C = 15h + 20$
$\displaystyle m = \frac{\Delta y}{\Delta x}$
2.6.5
Distance-Time Graphs
The slope is the speed. A flat stretch means stopped; a steeper line means faster; a falling line means heading back.
$\displaystyle \text{speed} = \text{slope} = \frac{\Delta d}{\Delta t}$
2.6.6
Speed-Time Graphs
The slope is the acceleration. A flat stretch means constant speed, not stopped.
$\displaystyle \text{acceleration} = \frac{\Delta v}{\Delta t}$
2.6.7
Calculating Distance From a Speed-Time Graph
Distance traveled is the area under the graph. Split it into rectangles and triangles.
e.g. 0 to 10 m/s over 4 s → $\tfrac12 \cdot 4 \cdot 10 = 20$ m
$\displaystyle d = \text{area under the } v\text{–}t \text{ graph}$

2.7 Systems of Equations

A system’s solution is where the lines meet. When elimination cancels both variables, the leftover statement tells you which case you have.

2.7.1
Introduction to the Elimination Method
Add or subtract the equations so that one variable cancels.
e.g. So $x = 7$ and $y = 3$
$\displaystyle \begin{array}{r} x + y = 10 \\ x - y = 4 \\ \hline 2x = 14 \end{array}$
2.7.2
Solving Systems of Linear Equations Using Elimination: One Transformation
Multiply one equation so a variable’s coefficients become opposites, then add.
e.g. $2x + y = 7$ , $x - 3y = 7$ : triple the first, add → $7x = 28$ , $x = 4$ , $y = -1$
2.7.3
Solving Systems of Linear Equations Using Elimination: Two Transformations
Multiply each equation by the other’s coefficient so one variable matches.
e.g. $3x + 2y = 16$ , $2x - 5y = -2$ : ×2 and ×3 → $19y = 38$ , $y = 2$ , $x = 4$
2.7.4
Systems of Linear Equations With Fractional Coefficients
Clear the fractions first: multiply each equation by its LCD.
$\displaystyle \tfrac{x}{2} + \tfrac{y}{3} = 4 \;\xrightarrow{\times 6}\; 3x + 2y = 24$
2.7.5
Systems of Linear Equations With Decimal Coefficients
Multiply by 10 or 100 to turn the decimals into integers.
$\displaystyle 0.2x + 0.5y = 1.6 \;\xrightarrow{\times 10}\; 2x + 5y = 16$
2.7.6
Systems of Equations With No Solutions and Infinitely Many Solutions
Both variables cancel. A false statement like $0 = 5$ means no solution (parallel lines). A true one like $0 = 0$ means infinitely many (the same line).
2.7.7
Consistency and Dependency in Linear Systems
Consistent: at least one solution. Inconsistent: none. Dependent: infinitely many. Independent: exactly one.
e.g. One solution is consistent and independent; the same line twice is consistent and dependent
2.7.8
Calculating the Intersection of Two Lines
Set the two expressions for $y$ equal, solve for $x$ , then find $y$ .
e.g. $y = 2x + 1$ and $y = -x + 7$ → $3x = 6$ → they meet at $(2, 5)$
$\displaystyle m_1x + b_1 = m_2x + b_2$
2.7.9
Modeling Using Systems of Linear Equations
Two unknowns need two equations. Usually one counts the items and one counts their value.
e.g. 150 tickets, adults $10, children $6, $1,140 → $a + c = 150$ , $10a + 6c = 1140$ → $a = 60$
2.7.10
Modeling Coin Problems Using Systems of Linear Equations
One equation for how many coins, one for what they’re worth. Work in cents.
e.g. → $d = 14$ dimes, $n = 16$ nickels
$\displaystyle n + d = 30$
$\displaystyle 5n + 10d = 220$
2.7.11
Solving Systems of Nonlinear Equations Using Graphs
Graph both. Each crossing point is a solution. A line and a parabola cross 0, 1, or 2 times.
2.7.12
Approximating Solutions to Systems of Linear Equations
Read the crossing off a graph, then check it by substituting. A graph reading is an estimate.
2.7.13
Approximating Solutions to Systems of Nonlinear Equations
Use a graph or a table to find where the two sides are closest, then narrow in.

2.8 Two-Variable Linear Inequalities

Draw the boundary line, decide dashed or solid, then use a test point to pick the side to shade.

2.8.1
Graphing Strict Two-Variable Linear Inequalities
For < or >, draw the boundary dashed, then shade the side that makes the inequality true. Test $(0, 0)$ if it isn’t on the line.
$\displaystyle y > mx + b$
$\displaystyle \to\; \text{dashed, shade above}$
2.8.2
Graphing Non-Strict Two-Variable Linear Inequalities
For ≤ or ≥, draw the boundary solid, because points on the line count.
$\displaystyle y \le mx + b$
$\displaystyle \to\; \text{solid, shade below}$
2.8.3
Further Graphing of Two-Variable Linear Inequalities
Rearrange into $y = \ldots$ first. Dividing by a negative flips the sign, which flips the shading.
$\displaystyle 2x - 3y > 6 \;\Rightarrow\; y < \tfrac{2}{3}x - 2$
2.8.4
Systems of Linear Inequalities
Graph each inequality. The solutions are where all the shadings overlap.
2.8.5
Modeling With Two-Variable Linear Inequalities
A budget or capacity limit becomes $ax + by \le C$ . Only sensible points count, often whole and nonnegative.
e.g. $60 for $5 books and $12 games: $5b + 12g \le 60$
2.8.6
Modeling With Systems of Linear Inequalities
Each constraint is one inequality. The workable choices lie in the overlap.
2.8.7
Modeling and Solving Systems of Linear Inequalities Using Technology
Enter each inequality in a graphing tool, read the overlap, and check its corner points.

Unit 3 13 topics

Units

Converting measurements, and quantities that vary together.

3.9 Working With Units

A conversion factor equals 1, so multiplying by it changes the units without changing the amount. The struck-through units cancel.

3.9.1
Unit Conversions Using Base Units of Mass
Metric prefixes are powers of ten: kilo is 1,000, centi is $\tfrac{1}{100}$ , milli is $\tfrac{1}{1000}$ . US: 16 oz = 1 lb, 2,000 lb = 1 ton.
e.g. $3.4\text{ kg} = 3400\text{ g}$
$\displaystyle 1\text{ kg} = 1000\text{ g}$
$\displaystyle 1\text{ g} = 1000\text{ mg}$
3.9.2
Unit Conversions Using Base Units of Length
1 km = 1,000 m and 1 m = 100 cm = 1,000 mm. US: 12 in = 1 ft, 3 ft = 1 yd, 5,280 ft = 1 mi.
e.g. $2.5\text{ km} = 2500\text{ m}$
3.9.3
Unit Conversions Using Units of Time
60 s = 1 min, 60 min = 1 h, 24 h = 1 day, 7 days = 1 week.
$\displaystyle 1\text{ h} = 3600\text{ s}$
3.9.4
Two-Step Unit Conversions
Multiply by conversion fractions arranged so the unwanted units cancel.
$\displaystyle \frac{60\text{ mi}}{1\text{ h}} \cdot \frac{5280\text{ ft}}{1\text{ mi}} \cdot \frac{1\text{ h}}{3600\text{ s}} = 88\ \frac{\text{ft}}{\text{s}}$
3.9.5
Converting Units of Area to Smaller Units
Square the length factor. Going to a smaller unit, multiply.
$\displaystyle 1\text{ m}^2 = 100^2\text{ cm}^2 = 10{,}000\text{ cm}^2$
3.9.6
Converting Units of Area to Larger Units
Square the length factor and divide.
$\displaystyle 5000\text{ cm}^2 = \frac{5000}{10{,}000}\text{ m}^2 = 0.5\text{ m}^2$
3.9.7
Converting Between Mixed Units
Convert everything to the smallest unit, compute, then convert back.
e.g. 2 h 45 min $= 165$ min; 200 min $=$ 3 h 20 min
3.9.8
Determining Units in Formulas
Units follow the same algebra as the numbers: they multiply and divide.
$\displaystyle \text{m} \times \text{m} = \text{m}^2$
$\displaystyle \frac{\text{m}}{\text{s}} \div \text{s} = \frac{\text{m}}{\text{s}^2}$
3.9.9
Selecting Units for Rates of Change
A rate’s unit is the $y$ -unit per $x$ -unit. Choose a scale that gives sensible numbers.
e.g. Fingernail growth in mm per week, not km per year
$\displaystyle \text{rate unit} = \frac{y\text{-unit}}{x\text{-unit}}$
3.9.10
Degrees of Accuracy
A measurement rounded to some unit can be off by up to half that unit either way.
$\displaystyle 7\text{ cm (to the nearest cm)}$
$\displaystyle \Rightarrow\; 6.5 \le x < 7.5$

3.10 Direct and Inverse Variation

Direct variation is a line through the origin, with y ÷ x constant. Inverse variation is a curve, with x × y constant.

3.10.1
Modeling with Direct Variation
$y$ varies directly with $x$ when $y = kx$ . Find $k$ from one pair of values, then use it.
e.g. $y = 12$ when $x = 3$ → $k = 4$ → at $x = 5$ , $y = 20$
$\displaystyle y = kx,\quad k = \frac{y}{x}$
3.10.2
Inverse Variation
When one quantity doubles, the other halves. Their product stays constant.
$\displaystyle y = \frac{k}{x} \quad\Leftrightarrow\quad xy = k$
3.10.3
Modeling With Inverse Variation
Find $k = xy$ from one pair, then $y = \tfrac{k}{x}$ .
e.g. 4 workers take 6 days → $k = 24$ → 8 workers take 3 days
$\displaystyle k = xy$

Unit 4 17 topics

Radicals and Rationals

Roots with variables, and fractions with variables.

4.11 Radical Expressions

A root is a fractional power. Even roots need an absolute value when the sign of x is unknown; odd roots do not.

4.11.1
Writing Radical Expressions Using Fractional Exponents
The denominator is the root; the numerator is the power.
e.g. $x^{3/2} = \sqrt{x^3}$
$\displaystyle a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$
4.11.2
The Square Root of a Perfect Square With Algebraic Expressions
Halve each exponent.
$\displaystyle \sqrt{a^{2n}} = a^n$
$\displaystyle \sqrt{x^2y^4} = xy^2 \quad (x \ge 0)$
4.11.3
The Square Root of a Perfect Square With Domain Restrictions
With no sign given, $\sqrt{x^2}$ is the distance $\lvert x \rvert$ , because a square root is never negative.
e.g. $\sqrt{(-5)^2} = 5$ , not $-5$
$\displaystyle \sqrt{x^2} = \lvert x \rvert$
$\displaystyle \sqrt{x^6} = \lvert x^3 \rvert$
4.11.4
The Cube Root of a Perfect Cube With Algebraic Expressions
Divide each exponent by 3. No absolute value: a cube root keeps the sign.
$\displaystyle \sqrt[3]{x^3} = x$
$\displaystyle \sqrt[3]{-8a^6} = -2a^2$
4.11.5
Simplifying Square Root Expressions Using the Product Rule
Split off the largest perfect square, including even powers of the variables.
$\displaystyle \sqrt{ab} = \sqrt{a}\,\sqrt{b}$
$\displaystyle \sqrt{50x^3} = \sqrt{25x^2 \cdot 2x} = 5x\sqrt{2x}$
4.11.6
Combining Radical Expressions Using the Product Rule
Multiply the outside numbers together and the insides together, then simplify.
e.g. $2\sqrt3 \cdot 5\sqrt6 = 10\sqrt{18} = 30\sqrt2$
$\displaystyle a\sqrt{b} \cdot c\sqrt{d} = ac\sqrt{bd}$
4.11.7
Simplifying Square Root Expressions Using the Quotient Rule
Split the root over top and bottom, or simplify the fraction inside first.
e.g. $\sqrt{\tfrac{18x^3}{2x}} = \sqrt{9x^2} = 3x$ for $x > 0$
$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
4.11.8
Evaluating Algebraic Radical Expressions
Substitute, simplify everything under the root, then take the root.
e.g. $\sqrt{b^2 - 4ac}$ at $a = 1, b = 5, c = 6$ → $\sqrt{1} = 1$
4.11.9
Adding and Subtracting Radical Expressions
Only like radicals combine: same index, same radicand. Simplify each one first.
$\displaystyle 3\sqrt{x} + 5\sqrt{x} = 8\sqrt{x}$
$\displaystyle \sqrt{12} + \sqrt{27} = 2\sqrt3 + 3\sqrt3 = 5\sqrt3$

4.12 Rational Expressions

4.12.1
Equivalent Expressions With Fractions
Multiply or divide top and bottom by the same nonzero expression.
$\displaystyle \frac{a}{b} = \frac{ak}{bk} \quad (k \ne 0)$
4.12.2
Simplifying Rational Expressions
Cancel common factors, never terms. Values that made the original denominator zero stay excluded.
$\displaystyle \frac{6x^2}{9x} = \frac{2x}{3},\ x \ne 0$
$\displaystyle \frac{x + 2}{x} \ne 2$
4.12.3
Simplifying Rational Expressions by Factoring
Factor top and bottom completely, then cancel the shared factors.
$\displaystyle \frac{x^2 - 9}{x^2 + 3x} = \frac{(x - 3)(x + 3)}{x(x + 3)} = \frac{x - 3}{x}$
4.12.4
Splitting Rational Expressions Into Separate Terms
A single denominator can be split under each term on top. The reverse fails: a sum in the denominator can’t be split.
e.g. $\tfrac{6x^2 + 3x}{3x} = 2x + 1$ for $x \ne 0$
$\displaystyle \frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$
$\displaystyle \frac{c}{a + b} \ne \frac{c}{a} + \frac{c}{b}$

4.13 Rational Equations

Cross-multiplying is multiplying both sides by both denominators at once. It only works when each side is a single fraction.

4.13.1
Solving Rational Equations Containing One Fractional Term
Isolate the fraction, then multiply both sides by its denominator. Exclude values that make it zero.
e.g. $\tfrac6x + 1 = 4 \Rightarrow \tfrac6x = 3 \Rightarrow x = 2$
4.13.2
Solving Rational Equations Using Cross-Multiplication
When one fraction equals another, multiply diagonally.
$\displaystyle \frac{a}{b} = \frac{c}{d} \;\Rightarrow\; ad = bc$
4.13.3
Solving Rational Equations Containing Binomials Using Cross-Multiplication
Cross-multiply, distribute both sides, solve. Check that no denominator becomes zero.
e.g. $x = 8$
$\displaystyle \frac{x + 1}{3} = \frac{x - 2}{2}$
$\displaystyle \Rightarrow\; 2x + 2 = 3x - 6$
4.13.4
Solving Rational Equations Using the Flip Method
If two fractions are equal, their reciprocals are too. Handy when the variable is in a denominator.
e.g. $\tfrac{3}{x - 1} = \tfrac14 \Rightarrow \tfrac{x - 1}{3} = 4 \Rightarrow x = 13$
$\displaystyle \frac{a}{b} = \frac{c}{d} \;\Rightarrow\; \frac{b}{a} = \frac{d}{c}$

Unit 5 14 topics

Functions

Rules that turn each input into exactly one output.

5.14 Functions

A function sends each input to exactly one output. On a graph, a vertical line that hits twice means some input has two outputs.

5.14.1
Introduction to Functions
A function gives each input exactly one output. $f(x)$ , read “f of x,” is the output at input $x$ .
$\displaystyle f(x) = 2x + 1 \;\Rightarrow\; f(3) = 7$
5.14.2
Visual Representations of Functions
Mapping diagrams, tables, and graphs. It is a function if no input has two arrows, two rows, or two points.
5.14.3
Graphs of Functions
The graph is every point $(x, f(x))$ . The value $f(a)$ is the height of the graph above $x = a$ .
5.14.4
The Domain of a Function
The allowed inputs. Exclude anything that divides by zero or takes an even root of a negative.
$\displaystyle f(x) = \frac{1}{x - 2}:\ x \ne 2$
$\displaystyle g(x) = \sqrt{x + 5}:\ x \ge -5$
5.14.5
The Vertical Line Test
A graph is a function if no vertical line crosses it more than once.
e.g. A circle fails; the parabola $y = x^2$ passes
5.14.6
Global Extrema of Functions
The highest output anywhere is the global maximum; the lowest is the global minimum. Some functions have neither.
5.14.7
End Behavior of Functions
What $f(x)$ does as $x \to \infty$ and as $x \to -\infty$ : rises, falls, or levels off.
e.g. $f(x) = -x^2$ falls at both ends
5.14.8
The Range of a Function
The outputs the function actually produces. Read them off the graph’s lowest and highest points.
e.g. Range of $x^2 + 1$ : $[1, \infty)$
5.14.9
The Range of a Function: Advanced Cases
Watch for values the graph approaches but never reaches, and for gaps.
e.g. $f(x) = \tfrac1x$ : range $(-\infty, 0) \cup (0, \infty)$
5.14.10
The Roots of a Function
Inputs where $f(x) = 0$ : where the graph crosses or touches the $x$ -axis.
e.g. Roots of $x^2 - 4$ : $\pm 2$
$\displaystyle f(x) = 0$
5.14.11
Increasing and Decreasing Functions
Increasing where the graph rises left to right, decreasing where it falls. State the $x$ -interval.
e.g. $x^2$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$
5.14.12
Piecewise Functions
Different rules on different parts of the domain. Find which piece the input belongs to, then use that rule.
e.g. $f(-3) = -1$ , $f(3) = 9$
$\displaystyle f(x) = \begin{cases} x + 2 & x < 0 \\ x^2 & x \ge 0 \end{cases}$
5.14.13
Modeling With Linear Functions
A constant rate of change means a linear model, $f(x) = mx + b$ .
5.14.14
Constructing Linear Functions
Find the slope from two points, then solve for $b$ .
e.g. $f(1) = 5$ , $f(3) = 11$ → $m = 3$ , $f(x) = 3x + 2$
$\displaystyle m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Unit 6 6 topics

Absolute Value

Distance from zero, in expressions, equations, and inequalities.

6.15 Absolute Value Expressions, Equations & Inequalities

|x − 3| is the distance from x to 3. Less than means close to 3; greater than means far from it.

6.15.1
Absolute Value Expressions
Absolute value is distance from zero, so it is never negative. Evaluate inside first.
e.g. $\lvert 3 - 8 \rvert = 5$
$\displaystyle \lvert x \rvert = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$
6.15.2
Rules of Absolute Value
It passes through multiplication and division, but not through addition.
$\displaystyle \lvert ab \rvert = \lvert a \rvert\,\lvert b \rvert$
$\displaystyle \left\lvert \tfrac{a}{b} \right\rvert = \tfrac{\lvert a \rvert}{\lvert b \rvert}$
$\displaystyle \lvert a + b \rvert \le \lvert a \rvert + \lvert b \rvert$
6.15.3
Further Rules of Absolute Value
A number and its opposite have the same absolute value.
$\displaystyle \lvert -a \rvert = \lvert a \rvert$
$\displaystyle \lvert a \rvert^2 = a^2$
$\displaystyle \sqrt{a^2} = \lvert a \rvert$
6.15.4
Absolute Value Equations
Isolate the absolute value, then split into two equations. A negative right side has no solution.
e.g. $\lvert 2x - 1 \rvert = 7 \Rightarrow x = 4$ or $-3$
$\displaystyle \lvert A \rvert = k,\ k \ge 0$
$\displaystyle \Rightarrow\; A = k \text{ or } A = -k$
6.15.5
Further Absolute Value Equations
When two absolute values are equal, the insides are equal or opposite.
$\displaystyle \lvert A \rvert = \lvert B \rvert$
$\displaystyle \Rightarrow\; A = B \text{ or } A = -B$
6.15.6
Absolute Value Inequalities
Less than means between, an “and.” Greater than means outside, an “or.”
$\displaystyle \lvert A \rvert < k \;\Rightarrow\; -k < A < k$
$\displaystyle \lvert A \rvert > k \;\Rightarrow\; A < -k \text{ or } A > k$

Unit 7 21 topics

Exponents

Exponent rules, and quantities that grow or shrink by a fixed factor.

7.16 Rules of Exponents

7.16.1
The Product Rule for Exponents With Algebraic Expressions
Multiply the coefficients, add the exponents of matching bases.
e.g. $3x^2 \cdot 4x^5 = 12x^7$
$\displaystyle a^m \cdot a^n = a^{m+n}$
7.16.2
The Quotient Rule for Exponents With Algebraic Expressions
Divide the coefficients, subtract the exponents of matching bases.
e.g. $\tfrac{10x^6y}{2x^2y} = 5x^4$
$\displaystyle \frac{a^m}{a^n} = a^{m-n}$
7.16.3
The Power Rule for Exponents With Algebraic Expressions
A power of a power multiplies the exponents.
e.g. $(x^3)^4 = x^{12}$
$\displaystyle (a^m)^n = a^{mn}$
7.16.4
The Power of Product Rule With Algebraic Expressions
The outside power goes to every factor, including the coefficient and its sign.
e.g. $(-2x^3y)^2 = 4x^6y^2$
$\displaystyle (ab)^n = a^nb^n$
7.16.5
The Power of Quotient Rule With Algebraic Expressions
The outside power goes to the top and the bottom.
e.g. $\left(\tfrac{x^2}{3}\right)^3 = \tfrac{x^6}{27}$
$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
7.16.6
Combining the Rules of Exponents With Algebraic Expressions
Apply outside powers, combine matching bases, and finish with positive exponents.
e.g. $\dfrac{(2x^2)^3}{4x^{-1}} = \dfrac{8x^6}{4x^{-1}} = 2x^7$

7.17 Exponential Expressions and Equations

7.17.1
Solving Exponential Equations
Rewrite both sides with the same base, then set the exponents equal.
e.g. $2^{x+1} = 32 = 2^5 \Rightarrow x = 4$
$\displaystyle b^m = b^n \;\Rightarrow\; m = n$
7.17.2
Solving Exponential Equations with Fractional Solutions
Use a base that both sides are powers of.
$\displaystyle 4^x = 8 \;\Rightarrow\; 2^{2x} = 2^3 \;\Rightarrow\; x = \tfrac{3}{2}$
7.17.3
Creating Exponential Growth Expressions
Growth by rate $r$ multiplies by $1 + r$ each period.
e.g. $500 growing 4% a year: $500(1.04)^t$
$\displaystyle a(1 + r)^t$
7.17.4
Creating Exponential Decay Expressions
Decay by rate $r$ multiplies by $1 - r$ each period.
e.g. A $20,000 car losing 15% a year: $20{,}000(0.85)^t$
$\displaystyle a(1 - r)^t$

7.18 Exponential Functions

Equal differences mean linear; equal ratios mean exponential. Any exponential growth eventually passes any linear growth.

7.18.1
Exponential Functions
Starting value $a$ , multiplied by base $b$ at each step. Growth if $b > 1$ , decay if $0 < b < 1$ .
$\displaystyle f(x) = a\,b^x \quad (a \ne 0,\ b > 0,\ b \ne 1)$
7.18.2
Modeling Exponential Growth With Functions
Find the starting amount and the growth factor per time unit. For doubling every $d$ units, use $2^{t/d}$ .
$\displaystyle P(t) = P_0(1 + r)^t$
$\displaystyle P(t) = P_0 \cdot 2^{t/d}$
7.18.3
Interpreting Exponential Growth
In $a(1.06)^t$ , $a$ is the starting value and $1.06$ means 6% growth each period.
e.g. $b = 1.25$ → 25% growth per period
$\displaystyle r = b - 1$
7.18.4
Solving Exponential Growth Problems
Substitute $t$ to find a future value. To find $t$ , match bases or try values.
e.g. 3,000 bacteria doubling hourly → after 4 h, $3000 \cdot 2^4 = 48{,}000$
7.18.5
Modeling Exponential Decay With Functions
Use the decay factor per period, or a half-life $h$ .
$\displaystyle P(t) = P_0(1 - r)^t$
$\displaystyle P(t) = P_0\left(\tfrac{1}{2}\right)^{t/h}$
7.18.6
Interpreting Exponential Decay
$b = 0.9$ means 90% is kept each period, so 10% is lost.
$\displaystyle r = 1 - b$
7.18.7
Solving Exponential Decay Problems
Count the periods, then apply the factor that many times.
e.g. 80 mg with a 6 h half-life → after 18 h, $80\left(\tfrac12\right)^3 = 10$ mg
7.18.8
Linear vs. Exponential Growth and Decay
Linear adds the same amount each step. Exponential multiplies by the same factor.
$\displaystyle \text{linear: } y_{n+1} - y_n = m$
$\displaystyle \text{exponential: } \frac{y_{n+1}}{y_n} = b$
7.18.9
Linear vs. Exponential Growth and Decay Models
In a table, constant differences mean linear and constant ratios mean exponential. Exponential growth eventually passes any linear growth.
e.g. 3, 6, 12, 24: ratio 2 → exponential
7.18.10
Graphing Exponential Growth Functions
Passes through $(0, a)$ , rises ever faster to the right, and flattens toward $y = 0$ on the left without touching it.
$\displaystyle y = a\,b^x,\ b > 1$
7.18.11
Graphing Exponential Decay Functions
Passes through $(0, a)$ and falls toward $y = 0$ on the right without reaching it.
$\displaystyle y = a\,b^x,\ 0 < b < 1$

Unit 8 20 topics

Polynomials

Building, multiplying, and taking apart expressions in powers of x.

8.19 Polynomials

A square with side a + b splits into four pieces. Its area is not a² + b²; the two rectangles make up the difference.

8.19.1
Introduction to Polynomials
A sum of terms $ax^n$ with whole-number exponents. No variables in a denominator or under a root.
e.g. $\tfrac1x$ and $\sqrt{x}$ are not polynomial terms
$\displaystyle 3x^4 - x^2 + 7x - 5$
8.19.2
The Degree of a Polynomial
The highest exponent. For a term with several variables, add their exponents. Standard form lists terms from highest degree down.
$\displaystyle \deg(5x^3 - 2x + 1) = 3$
$\displaystyle \deg(4x^2y^3) = 5$
8.19.3
Simplifying Polynomials
Combine like terms: same variables raised to the same powers.
$\displaystyle 3x^2 + 5x - x^2 + 2x = 2x^2 + 7x$
8.19.4
The Distributive Law for Polynomials
Multiply the outside term by every term inside.
$\displaystyle 2x(x^2 - 3x + 4) = 2x^3 - 6x^2 + 8x$
8.19.5
Adding and Subtracting Polynomials
Combine like terms. When subtracting, flip the sign of every term in the second polynomial.
$\displaystyle (4x^2 + 3x) - (x^2 - 5x) = 3x^2 + 8x$
8.19.6
Monomials, Binomials and Trinomials
One term, two terms, three terms.
$\displaystyle 7x^3 \qquad x + 4 \qquad x^2 - 5x + 6$
8.19.7
Multiplying Binomials
Every term times every term: First, Outer, Inner, Last.
$\displaystyle (a + b)(c + d) = ac + ad + bc + bd$
$\displaystyle (x + 3)(x - 5) = x^2 - 2x - 15$
8.19.8
Multiplying Polynomials
Multiply each term of the first by each term of the second, then combine like terms.
$\displaystyle (x + 2)(x^2 - x + 3)$
$\displaystyle = x^3 + x^2 + x + 6$
8.19.9
Squaring Binomials
Square, twice the product, square. The middle term is the one people forget.
e.g. $(x + 4)^2 = x^2 + 8x + 16$ , not $x^2 + 16$
$\displaystyle (a + b)^2 = a^2 + 2ab + b^2$
$\displaystyle (a - b)^2 = a^2 - 2ab + b^2$
8.19.10
Expanding Binomials Using Pascal's Triangle
Row $n$ gives the coefficients of $(a + b)^n$ ; each entry is the sum of the two above it. Powers of $a$ fall as powers of $b$ rise.
e.g. Rows: 1 · 1 1 · 1 2 1 · 1 3 3 1 · 1 4 6 4 1
$\displaystyle (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
8.19.11
The Difference of Squares Formula
The middle terms cancel.
e.g. $(x + 7)(x - 7) = x^2 - 49$
$\displaystyle (a + b)(a - b) = a^2 - b^2$

8.20 Factoring Polynomials

Factoring runs the box backward: the corners hold x² and the constant, the two middle cells add to the middle term, and the edges are the factors.

8.20.1
The Greatest Common Factor of Two Monomials
GCF of the coefficients, times each shared variable at its lowest power.
$\displaystyle \text{GCF}(12x^3y,\ 18x^2y^4) = 6x^2y$
8.20.2
Factoring Polynomials Using GCFs
Always look for a common factor first.
$\displaystyle 6x^3 - 9x^2 = 3x^2(2x - 3)$
8.20.3
Factoring Perfect Square Trinomials
First and last terms are squares and the middle is twice their product.
$\displaystyle a^2 + 2ab + b^2 = (a + b)^2$
$\displaystyle x^2 - 10x + 25 = (x - 5)^2$
8.20.4
Factoring Perfect Square Trinomials With Leading Coefficients
The same test, with a coefficient on the first term.
$\displaystyle 4x^2 + 12x + 9 = (2x + 3)^2$
8.20.5
Factoring Differences of Squares
Two squares with a minus between them.
e.g. A sum of squares $a^2 + b^2$ doesn’t factor
$\displaystyle a^2 - b^2 = (a + b)(a - b)$
$\displaystyle 9x^2 - 16 = (3x + 4)(3x - 4)$
8.20.6
Factoring Trinomials
For $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ .
e.g. $x^2 + 7x + 12 = (x + 3)(x + 4)$
$\displaystyle x^2 + bx + c = (x + p)(x + q)$
$\displaystyle pq = c,\quad p + q = b$
8.20.7
Factoring Trinomials Using Common Factors
Pull out the GCF first, then factor what is left.
$\displaystyle 2x^2 - 8x - 24 = 2(x^2 - 4x - 12)$
$\displaystyle = 2(x - 6)(x + 2)$
8.20.8
Factoring Trinomials With Leading Coefficients
For $ax^2 + bx + c$ , find two numbers that multiply to $ac$ and add to $b$ . Split the middle term and factor by grouping.
e.g. $2x^2 + 7x + 3$ : $ac = 6$ → 6 and 1 → $2x^2 + 6x + x + 3 = (2x + 1)(x + 3)$
8.20.9
Further Factoring Trinomials With Leading Coefficients
With a negative leading coefficient, factor out $-1$ first. Check by multiplying back.
e.g. $-3x^2 + x + 2 = -(3x^2 - x - 2) = -(3x + 2)(x - 1)$

Unit 9 12 topics

Sequences

Ordered lists built by a rule, especially adding the same amount each time.

9.21 Introduction to Sequences

9.21.1
Introduction to Sequences
An ordered list of terms. $a_n$ is the $n$ th term. An explicit formula gives any term straight from $n$ .
$\displaystyle a_n = n^2 \;\Rightarrow\; 1, 4, 9, 16, \ldots$
9.21.2
Recursive Sequences
Each term is built from the one before, starting from a given first term.
e.g. 3, 7, 15, 31, …
$\displaystyle a_1 = 3,\quad a_n = 2a_{n-1} + 1$
9.21.3
Fibonacci Sequences
Each term is the sum of the two before it.
e.g. 1, 1, 2, 3, 5, 8, 13, …
$\displaystyle F_n = F_{n-1} + F_{n-2},\quad F_1 = F_2 = 1$

9.22 Arithmetic Sequences

9.22.1
Arithmetic Sequences
Add the same common difference $d$ each time.
e.g. 5, 8, 11, 14: $d = 3$
$\displaystyle d = a_{n+1} - a_n$
9.22.2
Recursive Formulas for Arithmetic Sequences
Give the first term and the rule for the next.
$\displaystyle a_1 = \text{first term},\quad a_n = a_{n-1} + d$
9.22.3
The Nth Term of an Arithmetic Sequence
Start at $a_1$ and take $n - 1$ steps of $d$ .
e.g. 5, 8, 11, … → $a_{20} = 5 + 19 \cdot 3 = 62$
$\displaystyle a_n = a_1 + (n - 1)d$
9.22.4
Translating Between Explicit and Recursive Formulas for Arithmetic Sequences
Both use the same $a_1$ and $d$ .
$\displaystyle a_n = a_{n-1} + d$
$\displaystyle \leftrightarrow\; a_n = a_1 + (n - 1)d$
9.22.5
Finding the Common Difference of an Arithmetic Sequence
Subtract any term from the next. From two terms further apart, divide the gap by the number of steps.
$\displaystyle d = \frac{a_m - a_k}{m - k}$
9.22.6
Finding the Nth Term of an Arithmetic Sequence Given Two Terms
Find $d$ , step back to $a_1$ , then write the formula.
e.g. $a_3 = 10$ , $a_7 = 22$ → $d = 3$ , $a_1 = 4$ , $a_n = 3n + 1$
9.22.7
Determining Indexes of Terms in Arithmetic Sequences
Set the formula equal to the value and solve for $n$ . If $n$ isn’t a positive whole number, the value isn’t in the sequence.
e.g. Is 100 in $a_n = 3n + 1$ ? $n = 33$ , yes
9.22.8
Solving for Variables in Arithmetic Sequences
Three terms in a row are arithmetic when the middle one is the average of the other two.
e.g. $x,\ 2x + 1,\ 5x - 3$ → $2(2x + 1) = 6x - 3$ → $x = \tfrac52$
$\displaystyle b = \frac{a + c}{2}$
9.22.9
Modeling With Arithmetic Sequences
A fixed amount added per step is arithmetic. Check whether counting starts at $n = 0$ or $n = 1$ .
e.g. 20 seats in row 1, 2 more per row → row 15 has $20 + 14 \cdot 2 = 48$

Unit 10 60 topics

Geometry

Angle proofs, congruent triangles, transformations, right triangles, circles, and coordinates.

10.23 Geometric Proofs

The angle numbers used in the proofs below. Highlighted: 4 and 5, a pair of alternate interior angles.

10.23.1
Identifying Angle Postulates
A postulate is accepted without proof and used as a starting point. The ones used most: angle addition, linear pairs, and corresponding angles.
e.g. A proof is a list of statements, each with a reason: given, a definition, a postulate, or a theorem already proved
$\displaystyle m\angle ABD = m\angle ABC + m\angle CBD$
$\displaystyle \text{linear pair: } m\angle 1 + m\angle 2 = 180^\circ$
$\displaystyle \ell \parallel m \;\Rightarrow\; \text{corresponding angles} \cong$
10.23.2
Identifying Angle Theorems
A theorem is proved from postulates and earlier theorems. Vertical angles are congruent. With parallel lines, alternate angles are congruent and consecutive interior angles are supplementary. Complements or supplements of congruent angles are congruent.
e.g. In the diagram: $\angle 1 \cong \angle 4$ (vertical), $\angle 4 \cong \angle 5$ (alternate interior)
10.23.3
Proving Alternate Angle Theorems
To prove alternate interior angles congruent when the lines are parallel, go through a corresponding pair and a vertical pair, then chain them.
$\displaystyle \angle 4 \cong \angle 8 \quad \text{corresponding angles}$
$\displaystyle \angle 8 \cong \angle 5 \quad \text{vertical angles}$
$\displaystyle \therefore\ \angle 4 \cong \angle 5 \quad \text{transitive property}$
10.23.4
Further Proving Alternate Angle Theorems
The converse: if alternate angles are congruent, the lines are parallel. Turn the alternate pair into a corresponding pair using vertical angles, then use the converse of the corresponding angles postulate.
$\displaystyle \angle 4 \cong \angle 5 \;\Rightarrow\; \ell \parallel m$
10.23.5
Proving Consecutive Angle Theorems
To prove consecutive interior angles supplementary when the lines are parallel, combine a corresponding pair with a linear pair and substitute.
$\displaystyle \angle 4 \cong \angle 8 \quad \text{corresponding angles}$
$\displaystyle m\angle 6 + m\angle 8 = 180^\circ \quad \text{linear pair}$
$\displaystyle \therefore\ m\angle 6 + m\angle 4 = 180^\circ \quad \text{substitution}$
10.23.6
Further Proving Consecutive Angle Theorems
The converse: if consecutive interior angles are supplementary, the lines are parallel. Use a linear pair to show a corresponding pair is congruent.
$\displaystyle m\angle 4 + m\angle 6 = 180^\circ \;\Rightarrow\; \ell \parallel m$
10.23.7
Proving Perpendicular Line Theorems
If two lines form a linear pair of congruent angles, each angle is 90°, so the lines are perpendicular. If two lines meet at one right angle, all four angles are right angles.
$\displaystyle x + x = 180^\circ \;\Rightarrow\; x = 90^\circ$
10.23.8
Further Proving Perpendicular Line Theorems
A line perpendicular to one of two parallel lines is perpendicular to the other. Two lines perpendicular to the same line are parallel. Both follow from corresponding angles.
$\displaystyle t \perp \ell,\ \ell \parallel m \;\Rightarrow\; t \perp m$
$\displaystyle \ell \perp t,\ m \perp t \;\Rightarrow\; \ell \parallel m$

10.24 Geometric Transformations

Dashed: the original. Solid: the image. Translations, reflections, and rotations keep size and shape. A dilation keeps the shape and changes the size. A reflection reverses orientation.

10.24.1
Translations of Geometric Figures
Slide every point the same distance in the same direction. Size, shape, and orientation stay the same.
e.g. Translate $(2, 1)$ by $(3, -4)$ → $(5, -3)$
$\displaystyle (x, y) \to (x + a,\ y + b)$
10.24.2
Rotations of Geometric Figures
Turn every point through the same angle about a center. Counterclockwise is the positive direction. Size and shape stay the same.
e.g. A 90° rotation carries each point a quarter of the way around the center
10.24.3
Rotating Objects in the Coordinate Plane Using Functions
Counterclockwise rotations about the origin.
e.g. $(3, 1)$ rotated 90° → $(-1, 3)$
$\displaystyle 90^\circ:\ (x, y) \to (-y,\ x)$
$\displaystyle 180^\circ:\ (x, y) \to (-x,\ -y)$
$\displaystyle 270^\circ:\ (x, y) \to (y,\ -x)$
10.24.4
Reflections of Geometric Figures in the Cartesian Plane
Flip across a line: each point lands the same distance away on the other side. Orientation reverses.
$\displaystyle x\text{-axis}:\ (x, y) \to (x,\ -y)$
$\displaystyle y\text{-axis}:\ (x, y) \to (-x,\ y)$
$\displaystyle y = x:\ (x, y) \to (y,\ x)$
10.24.5
Reflections of Figures Across Arbitrary Lines
The mirror line is the perpendicular bisector of each point and its image. For a vertical or horizontal line, measure across and go the same distance past.
e.g. Reflect $(1, 4)$ in $x = 3$ → $(5, 4)$
$\displaystyle x = k:\ (x, y) \to (2k - x,\ y)$
$\displaystyle y = k:\ (x, y) \to (x,\ 2k - y)$
10.24.6
Dilations of Geometric Figures
Scale every distance from a center by a factor $k$ . $k > 1$ enlarges, $0 < k < 1$ shrinks. Angles stay the same; lengths multiply by $k$ and areas by $k^2$ .
$\displaystyle \text{new length} = k \times \text{old}$
$\displaystyle \text{new area} = k^2 \times \text{old}$
10.24.7
Dilations of Figures in the Coordinate Plane
Centered at the origin, multiply both coordinates by $k$ . From another center, scale each point’s offset from that center.
e.g. Dilate $(2, -3)$ by 3 about the origin → $(6, -9)$
$\displaystyle (x, y) \to (kx,\ ky)$
$\displaystyle \text{about } (a, b):\ x' = a + k(x - a)$
$\displaystyle \phantom{\text{about } (a, b):\ } y' = b + k(y - b)$
10.24.8
Stretches of Geometric Figures
Scale in one direction only, so the shape changes: a square becomes a rectangle, a circle an oval.
$\displaystyle \text{horizontal by } k:\ (x, y) \to (kx,\ y)$
$\displaystyle \text{vertical by } k:\ (x, y) \to (x,\ ky)$
10.24.9
Combining Stretches of Geometric Figures
Stretching by $a$ horizontally and $b$ vertically multiplies areas by $ab$ . Equal stretches make a dilation.
$\displaystyle (x, y) \to (ax,\ by)$
$\displaystyle \text{area} \times ab$
10.24.10
Combining Geometric Transformations
Apply them in order; the order usually matters. Translations, rotations, and reflections keep size and shape (congruence). Add a dilation and only the shape is kept (similarity).
e.g. Translate right 2, then reflect in the $y$ -axis: $(1, 1) \to (3, 1) \to (-3, 1)$ . In the other order it ends at $(1, 1)$
10.24.11
Reflective Symmetry
A line of symmetry is a mirror line that maps the shape onto itself. A regular $n$ -gon has $n$ of them.
e.g. A rectangle has 2 lines of symmetry; a square has 4
10.24.12
Rotational Symmetry
A shape has rotational symmetry when a turn of less than 360° maps it onto itself. Its order is how many times it fits in one full turn.
e.g. A regular hexagon has order 6: it fits every 60°
$\displaystyle \text{smallest angle} = \frac{360^\circ}{\text{order}}$

10.25 Congruence

Thick sides and shaded angles are the parts that must match. Any one of these five is enough to prove two triangles congruent. SSA and AAA are not.

10.25.1
The ASA Congruence Criterion
Two angles and the side between them match, so the triangles are congruent.
$\displaystyle \angle A \cong \angle D,\quad \overline{AB} \cong \overline{DE},\quad \angle B \cong \angle E$
$\displaystyle \Rightarrow\; \triangle ABC \cong \triangle DEF$
10.25.2
The AAS Congruence Criterion
Two angles and a side that is not between them. It works because the third angles must also match (the angles sum to 180°), which turns it into ASA.
10.25.3
The SAS Congruence Criterion
Two sides and the angle between them. The angle must be the one the two sides form.
$\displaystyle \overline{AB} \cong \overline{DE},\quad \angle A \cong \angle D,\quad \overline{AC} \cong \overline{DF}$
10.25.4
The SSS Congruence Criterion
All three pairs of sides match. Three side lengths fix a triangle’s shape completely.
10.25.5
The HL Congruence Criterion
Right triangles only: the hypotenuse and one leg match. The Pythagorean theorem forces the other legs to match too.
10.25.6
Combining Congruence Criteria for Triangles
List what is given, add what you can deduce (a shared side, vertical angles, a third angle), then pick the criterion that fits. SSA and AAA do not prove congruence.
e.g. A side shared by two triangles is congruent to itself (reflexive property)
10.25.7
Rigid Motions and Congruence
Two figures are congruent exactly when some sequence of translations, rotations, and reflections maps one onto the other. Then every matching part is congruent (CPCTC: corresponding parts of congruent triangles are congruent).
10.25.8
Properties of Congruence
Congruence is reflexive, symmetric, and transitive.
$\displaystyle \text{reflexive: } F \cong F$
$\displaystyle \text{symmetric: } F \cong G \;\Rightarrow\; G \cong F$
$\displaystyle \text{transitive: } F \cong G,\ G \cong H \;\Rightarrow\; F \cong H$
10.25.9
Proving Congruence Statements
Give each step a reason: the givens, the deduced facts, then the criterion. Vertex order matters: $\triangle ABC \cong \triangle DEF$ pairs $A$ with $D$ , $B$ with $E$ , and $C$ with $F$ . Once the triangles are congruent, CPCTC gives any matching part.

10.26 Triangles

Memorize these two shapes. Any 45°–45°–90° triangle has sides in the ratio 1 : 1 : √2; any 30°–60°–90° triangle, 1 : √3 : 2. The hypotenuse is always the longest side.

10.26.1
The Pythagorean Theorem
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the legs. It works in reverse too: if the sides fit, the angle is right.
e.g. Legs 5 and 12 → $c = \sqrt{169} = 13$
$\displaystyle a^2 + b^2 = c^2$
10.26.2
Isosceles Right Triangles
The 45°–45°–90° triangle. The legs are equal and the hypotenuse is a leg times $\sqrt2$ .
e.g. Leg 4 → hypotenuse $4\sqrt2$ ; hypotenuse 10 → legs $5\sqrt2$
$\displaystyle s : s : s\sqrt{2}$
10.26.3
30-60-90 Triangles
Half of an equilateral triangle. The short leg faces 30°; the hypotenuse is twice it; the long leg is $\sqrt3$ times it.
e.g. Short leg 5 → long leg $5\sqrt3$ , hypotenuse 10
$\displaystyle s : s\sqrt{3} : 2s$
10.26.4
The Area of an Isosceles Right Triangle
The two legs are the base and the height.
e.g. Leg 6 → $A = 18$
$\displaystyle A = \tfrac{1}{2}s^2$
$\displaystyle A = \tfrac{1}{4}h^2 \;\text{ from the hypotenuse } h$
10.26.5
The Area of a 30-60-90 Triangle
The two legs are the base and the height.
e.g. Short leg 2 → $A = 2\sqrt3$
$\displaystyle A = \tfrac{1}{2} \cdot s \cdot s\sqrt{3} = \frac{\sqrt{3}}{2}s^2$
10.26.6
The Area of an Equilateral Triangle
Its height splits it into two 30-60-90 triangles, so the height is $\tfrac{\sqrt3}{2}$ of a side.
e.g. Side 6 → $A = 9\sqrt3$
$\displaystyle h = \frac{\sqrt{3}}{2}s$
$\displaystyle A = \frac{\sqrt{3}}{4}s^2$
10.26.7
Diagonals of Squares
A diagonal cuts a square into two isosceles right triangles.
e.g. Side 7 → diagonal $7\sqrt2$
$\displaystyle d = s\sqrt{2}$
$\displaystyle A = \tfrac{1}{2}d^2$
10.26.8
Medians and Centroids of Triangles
A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid, $\tfrac23$ of the way along each median from its vertex. Its coordinates average the vertices’.
e.g. Vertices $(0, 0), (6, 0), (0, 9)$ → centroid $(2, 3)$
$\displaystyle G = \left(\frac{x_1 + x_2 + x_3}{3},\ \frac{y_1 + y_2 + y_3}{3}\right)$
10.26.9
The Exterior Angle Theorem
An exterior angle of a triangle equals the sum of the two interior angles not next to it.
e.g. Interior angles 50° and 65° → the exterior angle at the third vertex is 115°
$\displaystyle d = a + b$

10.27 Circles

An arc is part of the edge, a sector is a slice, and a segment is cut off by a chord. Arc length and sector area are the same fraction, θ/360°, of the whole.

10.27.1
Circles
Every point at a fixed distance, the radius, from the center. The diameter is twice the radius. A chord joins two points on the circle; a diameter is the longest chord.
$\displaystyle d = 2r$
10.27.2
Arcs, Segments, and Sectors of Circles
An arc is part of the edge. A sector is a slice bounded by two radii and an arc. A segment is the region between a chord and its arc. Minor arcs are under 180°, major arcs over.
10.27.3
The Circumference of a Circle
The distance around. $\pi \approx 3.14159$ is circumference divided by diameter for every circle.
e.g. $r = 5$ → $C = 10\pi \approx 31.4$
$\displaystyle C = 2\pi r = \pi d$
10.27.4
Areas of Circles
Square the radius, not the diameter.
e.g. $d = 10$ → $r = 5$ , $A = 25\pi$
$\displaystyle A = \pi r^2$
10.27.5
Central Angles and Arcs
A central angle has its vertex at the center. The arc it cuts off has the same degree measure.
$\displaystyle \text{arc measure} = \text{central angle}$
$\displaystyle \text{whole circle} = 360^\circ$
10.27.6
Calculating Arc Lengths of Circular Sectors
The arc is the angle’s share of the whole circumference.
e.g. $r = 6$ , $\theta = 60^\circ$ → $L = \tfrac16 \cdot 12\pi = 2\pi$
$\displaystyle L = \frac{\theta}{360^\circ} \cdot 2\pi r$
10.27.7
Calculating Areas of Circular Sectors
The sector is the angle’s share of the whole area.
e.g. $r = 6$ , $\theta = 60^\circ$ → $A = \tfrac16 \cdot 36\pi = 6\pi$
$\displaystyle A = \frac{\theta}{360^\circ} \cdot \pi r^2$
10.27.8
Further Calculating Areas of Sectors
Work backward from an area or arc length to the angle or radius. Arc length and sector area are linked.
e.g. Area $12\pi$ , $r = 6$ → $\tfrac{12\pi}{36\pi} = \tfrac13$ of the circle → $\theta = 120^\circ$
$\displaystyle A = \tfrac{1}{2}rL$
10.27.9
Problem Solving With Circles
Break the region into circles, sectors, and polygons, then add or subtract. A segment is a sector minus its triangle.
e.g. Quarter circle of radius 4: segment $= 4\pi - 8$
$\displaystyle A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$
10.27.10
Circles Inscribed in Squares
The circle touches all four sides, so its diameter equals the side of the square.
e.g. Side 10 → $r = 5$ , area $25\pi$
$\displaystyle d = s,\quad r = \tfrac{s}{2}$
$\displaystyle A_{\text{circle}} = \frac{\pi}{4}s^2$
10.27.11
Squares Inscribed in Circles
The square’s corners touch the circle, so its diagonal is a diameter.
e.g. $r = 5$ → side $5\sqrt2$ , area 50
$\displaystyle s\sqrt{2} = 2r \;\Rightarrow\; s = r\sqrt{2}$
$\displaystyle A_{\text{square}} = 2r^2$

10.28 Coordinate Geometry

The changes in x and y form a right triangle, so distance comes from the Pythagorean theorem. The midpoint averages the endpoints.

10.28.1
Parallel Lines in the Coordinate Plane
Parallel lines have the same slope and different $y$ -intercepts. All vertical lines are parallel to each other.
e.g. $y = 2x + 1$ and $y = 2x - 4$ are parallel
$\displaystyle m_1 = m_2$
10.28.2
Finding Equations of Parallel Lines
Keep the slope, then put the given point into point-slope form.
e.g. Parallel to $y = 3x - 2$ through $(1, 5)$ : $y = 3x + 2$
$\displaystyle y - y_1 = m(x - x_1)$
10.28.3
Perpendicular Lines in the Coordinate Plane
Perpendicular slopes are negative reciprocals: they multiply to $-1$ . Horizontal and vertical lines are perpendicular.
e.g. Slope $\tfrac23$ → perpendicular slope $-\tfrac32$
$\displaystyle m_1 m_2 = -1$
$\displaystyle m_2 = -\frac{1}{m_1}$
10.28.4
Finding Equations of Perpendicular Lines
Flip the slope and change its sign, then use the given point.
e.g. Perpendicular to $y = 2x + 1$ through $(4, 1)$ : $y = -\tfrac12x + 3$
$\displaystyle y - y_1 = -\frac{1}{m}(x - x_1)$
10.28.5
Midpoints in the Coordinate Plane
Average the $x$ -coordinates and average the $y$ -coordinates.
e.g. $(2, 7)$ and $(8, -1)$ → $(5, 3)$
$\displaystyle M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$
10.28.6
Partitioning Line Segments
The point a fraction $t$ of the way from $A$ to $B$ adds that fraction of each change. A ratio $m : n$ means $t = \tfrac{m}{m + n}$ .
e.g. $A(1, 2)$ to $B(7, 11)$ in ratio $1 : 2$ → $t = \tfrac13$ → $(3, 5)$
$\displaystyle P = \big(x_1 + t(x_2 - x_1),\ y_1 + t(y_2 - y_1)\big)$
10.28.7
The Distance Formula
The Pythagorean theorem applied to the horizontal and vertical changes.
e.g. $(1, 2)$ to $(4, 6)$ → $\sqrt{9 + 16} = 5$
$\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
10.28.8
The Shortest Distance Between a Point and a Line
Measure along the perpendicular: find the perpendicular through the point, intersect it with the line, then use the distance formula. For a line $Ax + By + C = 0$ there is a shortcut.
e.g. $(0, 0)$ to $3x + 4y - 10 = 0$ → $\tfrac{10}{5} = 2$
$\displaystyle d = \frac{\lvert Ax_0 + By_0 + C \rvert}{\sqrt{A^2 + B^2}}$
10.28.9
Calculating Perimeters in the Plane
Find each side with the distance formula and add. Horizontal and vertical sides are simple differences.
e.g. Triangle $(0, 0), (4, 0), (4, 3)$ : $4 + 3 + 5 = 12$
$\displaystyle P = \text{sum of the side lengths}$
10.28.10
Calculating Areas of Rectangles in the Plane
Find two adjacent sides with the distance formula and multiply. Check the corner is square: the slopes multiply to $-1$ .
e.g. $(0, 0), (4, 2), (3, 4), (-1, 2)$ : sides $\sqrt{20}$ and $\sqrt5$ → $A = 10$
$\displaystyle A = l \times w$
10.28.11
Calculating Areas of Triangles and Quadrilaterals in the Plane
Use a horizontal or vertical side as the base when you can. Otherwise box the shape in a rectangle and subtract the corner triangles, or use the formula below.
e.g. $(0, 0), (6, 0), (2, 5)$ : base 6, height 5 → $A = 15$
$\displaystyle A = \tfrac{1}{2}\big\lvert x_1(y_2 - y_3) + x_2(y_3 - y_1)$
$\displaystyle \qquad + x_3(y_1 - y_2)\big\rvert$

Unit 11 6 topics

Statistics

Two variables at once: plotting pairs, fitting a line, and what correlation can’t show.

11.29 Correlation

r runs from −1 to 1. The sign gives the direction; the closer to ±1, the tighter the points hug the trend line.

11.29.1
Scatter Plots
Plot paired data $(x, y)$ as points to look for a relationship. The explanatory variable goes on the $x$ -axis.
11.29.2
Trend Lines
A straight line through the middle of the cloud, with about as many points above it as below.
$\displaystyle \hat{y} = mx + b$
11.29.3
Making Predictions Using Trend Lines
Substitute $x$ into the trend line. Predicting inside the data’s range (interpolation) is safer than beyond it (extrapolation).
11.29.4
Interpreting Trend Line Coefficients
The slope is the predicted change in $y$ for each one-unit increase in $x$ . The intercept is the predicted $y$ at $x = 0$ , which may be meaningless.
e.g. $\hat{y} = 2.5x + 60$ , hours studied vs. score: each hour adds about 2.5 points
11.29.5
Linear Correlation
$r$ measures how tightly points hug a line, from $-1$ to $1$ . The sign gives direction; near 0 means no linear pattern. Correlation does not show cause.
$\displaystyle -1 \le r \le 1$
11.29.6
Correlation vs. Causation
A strong correlation shows two variables move together, not that one causes the other. A third variable may drive both, the cause may run the other way, or it may be chance. A controlled experiment is what isolates a cause.
e.g. Ice cream sales and drownings rise together because hot weather drives both

No topic matches that search. Try a single word, like “circle” or “slope.”

Topic numbers match the course outline, so a topic here lines up with the same number in the lesson list.

1.1 Linear Equations

Solving Linear Equations With Unknown Coefficients

Solving Linear Equations With Unknown Coefficients by Factoring

Solving Linear Inequalities With Unknown Parameters

Solving Many-Variable Equations

1.2 Modeling With Linear Equations

Interpreting Linear Expressions

Modeling With Linear Equations

Consecutive Integer Problems

Speed, Distance, Time Problems

Further Speed, Distance, Time Problems

Modeling Work Problems

Modeling Mixture Problems

1.3 Linear Inequalities

Solving Compound Inequalities

Interval Notation

Unbounded Intervals

Unions of Intervals

Intersections of Intervals

Compound OR Inequalities

Compound AND Inequalities

Introduction to Modeling With Inequalities

Modeling With One-Step Inequalities

Modeling With Two-Step Inequalities

Modeling With Compound Inequalities

1.4 Nonlinear Equations

Solving Equations Using the Square Root Method

Solving Equations With Odd Exponents Using the Nth Root Method

Solving Equations With Even Exponents Using the Nth Root Method

2.5 Graphs of Linear Equations

Equations of Lines in Slope-Intercept Form

Properties of Lines Given in Slope-Intercept Form

Equations of Lines in Point-Slope Form

Equations of Lines in Standard Form

Properties of Lines Given in Standard Form

2.6 Modeling With Two-Variable Linear Equations

Modeling With Linear Equations in Two Variables

Modeling Totals Using Two-Variable Linear Equations

Analyzing and Interpreting Graphs of Linear Equations

Further Modeling With Linear Equations in Two Variables

Distance-Time Graphs

Speed-Time Graphs

Calculating Distance From a Speed-Time Graph

2.7 Systems of Equations

Introduction to the Elimination Method

Solving Systems of Linear Equations Using Elimination: One Transformation

Solving Systems of Linear Equations Using Elimination: Two Transformations

Systems of Linear Equations With Fractional Coefficients

Systems of Linear Equations With Decimal Coefficients

Systems of Equations With No Solutions and Infinitely Many Solutions

Consistency and Dependency in Linear Systems

Calculating the Intersection of Two Lines

Modeling Using Systems of Linear Equations

Modeling Coin Problems Using Systems of Linear Equations

Solving Systems of Nonlinear Equations Using Graphs

Approximating Solutions to Systems of Linear Equations

Approximating Solutions to Systems of Nonlinear Equations

2.8 Two-Variable Linear Inequalities

Graphing Strict Two-Variable Linear Inequalities

Graphing Non-Strict Two-Variable Linear Inequalities

Further Graphing of Two-Variable Linear Inequalities

Systems of Linear Inequalities

Modeling With Two-Variable Linear Inequalities

Modeling With Systems of Linear Inequalities

Modeling and Solving Systems of Linear Inequalities Using Technology

3.9 Working With Units

Unit Conversions Using Base Units of Mass

Unit Conversions Using Base Units of Length

Unit Conversions Using Units of Time

Two-Step Unit Conversions

Converting Units of Area to Smaller Units

Converting Units of Area to Larger Units

Converting Between Mixed Units

Determining Units in Formulas

Selecting Units for Rates of Change

Degrees of Accuracy

3.10 Direct and Inverse Variation

Modeling with Direct Variation

Inverse Variation

Modeling With Inverse Variation