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Fall 2026

Math 130 — Test 1 Review

Chapter R through Section 2.5, plus maximum height from Section 3.1. Dates and deadlines are on the MATH 130 page.

What Test 1 covers

Chapter R through Section 2.5, plus maximum height

Test 1 covers Chapter R, Chapter 1, and Chapter 2 through Section 2.5. From Section 3.1 it includes only maximum height: find the vertex time with t = −b/(2a), then substitute to find the height. No formulas are provided on the test. Work through each step in order: learn with the slides, practice, then take the checkpoint. Dates are on the MATH 130 page.

Step 1 · Chapter R · Quiz 1

Algebra foundations: factor first

Intervals, exponent rules, factoring, and rational expressions.

Learn with Unit 1A
Step 2 · Sections 1.1 to 1.4 · Quiz 2

Equations, formulas, and roots

Fractional equations, formula rearrangement, mixtures, and solving quadratics.

Learn with Unit 1D
Step 3 · Sections 2.1 to 2.4 · Quiz 3

Coordinate geometry and quadratic applications

Distance, midpoint, slope, line equations, intercepts, circle form, and a quadratic application.

Learn with Unit 1C
Step 4 · Sections 2.4 and 2.5 · Quiz 4

Lines: parallel and perpendicular

Line equations and parallel and perpendicular lines, with review of earlier topics.

Lines review (PDF)

Ready to move on? Aim for at least 80% on each checkpoint and be able to explain every corrected error.

Topics for Test 1

Open a topic to review its key ideas and examples. Quizzes and Test 1 are on paper in class; ALEKS work is online.

Homework and test coverage differ. ALEKS assigns all of Sections 2.5 and 3.1. From Section 3.1, Test 1 includes only maximum height: find the vertex time with t = −b/(2a), then substitute to find the height. Other Section 3.1 topics are not on Test 1. No formulas are provided on the test.
Open the formula sheet

Key Concepts

  • Set-builder notation: e.g. \{x \mid x \geq 3\}
  • Interval notation: brackets [ ] include; parentheses ( ) exclude
  • Union A \cup B: elements in either set
  • Intersection A \cap B: elements in both sets

Common Intervals

[a, b] \quad (a, b) \quad [a, \infty) \quad (-\infty, b]
⚡ ∞ and −∞ always get parentheses, never brackets!
📺 Video Walkthroughs
How to Find the Intersection and Union of Two Intervals
Interval notation with number line visuals
Intersection and Union of Discrete Sets
Working with finite sets — listing elements in ∩ and ∪

Key Concepts

  • |a| = distance from a to 0 — always ≥ 0
  • |a| = a if a \geq 0; |a| = -a if a < 0
|x| = c \implies x = c \text{ or } x = -c \quad(c \geq 0)
⚡ Remember: \sqrt{x^2} = |x|
📺 Video Walkthrough
Challenging Absolute Value Equations
Quick walkthrough of tricky absolute value problems

Core Laws

  • a^m \cdot a^n = a^{m+n}
  • \frac{a^m}{a^n} = a^{m-n}
  • (a^m)^n = a^{mn}
  • (ab)^n = a^n b^n
  • a^{-n} = \frac{1}{a^n}
⚡ Write all final answers with positive exponents only.
📺 Video Walkthrough
Rational Exponents with a Few Examples
Converting between radicals and fractional exponents

Methods (always start with GCF!)

  • GCF — always factor out first
  • Grouping — split into pairs
  • Difference of squares: a^2 - b^2 = (a-b)(a+b)
  • Difference of cubes: a^3 - b^3 = (a-b)(a^2+ab+b^2)
  • Sum of cubes: a^3 + b^3 = (a+b)(a^2-ab+b^2)
  • Trinomials: find two numbers that multiply to ac and add to b
⚡ Mnemonic for cubes: "SOAP" — Same, Opposite, Always Positive.
📺 Video Walkthroughs
Factor a Quadratic with Lead Coefficient = 1
Finding two numbers that multiply and add correctly
How to Factor Quadratics with a Lead Coefficient
AC method for trinomials when a ≠ 1
How to Solve Quadratics by Factoring
GCF, trinomials without & with lead coefficients
  • Always factor first
  • Add/subtract: find LCD, rewrite, combine numerators
  • Multiply: factor → cancel → multiply
  • Divide: flip second fraction → multiply
⚡ Factor first! This is the #1 mistake on rational expression problems.
📺 Video Walkthrough
Solving Rational Equations
Finding LCD, clearing fractions, checking for extraneous solutions

Linear equations, formulas, and mixtures

  • With numerical fractions: multiply every term by the LCD
  • For a literal formula: undo operations while keeping grouped terms together
  • For a two-part mixture: define one amount as x, the other as T-x, and use c_1x+c_2(T-x)=c_T T

Quadratic equations

  • Factoring: write standard form, factor, then use the zero-product property
  • Completing the square: isolate the variable terms and add \left(\frac{b}{2}\right)^2 to both sides
  • Quadratic formula: identify a, b, and c, compute the discriminant, and simplify the exact roots
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Quiz 2 does not include rational equations with variable denominators or special-case linear classifications.
📺 Video Walkthroughs
How to Solve Quadratics by Factoring
GCF, trinomials without & with lead coefficients
Solving Rational Equations
ALEKS and Test 1 extension: denominator restrictions and extraneous solutions
\text{Midpoint} = \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
⚡ Round to one decimal place when instructed!
📺 Video Walkthrough
Midpoint & Distance Formula
Step-by-step examples with coordinate pairs
(x-h)^2 + (y-k)^2 = r^2

Center = (h,k), Radius = r

⚡ Right side is r², not r. Radius 5 → write 25.
📺 Video Walkthrough
Standard Equation of a Circle Formula Explained!
Center, radius, and writing the equation step by step
  • Slope-intercept: y = mx + b
  • Point-slope: y - y_1 = m(x - x_1)
  • General: Ax + By + C = 0
\text{Parallel: } m_1 = m_2 \qquad \text{Perp: } m_1 \cdot m_2 = -1
⚡ Perpendicular: flip and negate. Slope 3 → perp slope −1/3.
📺 Video Walkthroughs
Slope Intercept vs Point Slope Form
When to use each form and converting between them
Finding Slopes of Parallel and Perpendicular Lines
Identifying slopes and graphing parallel & perpendicular lines
x = -\frac{b}{2a}

Plug back into f(x) for the max/min value.

  • a > 0 → minimum
  • a < 0 → maximum
⚡ The question usually wants the y-value (height, etc.), not the x-value.
📺 Video Walkthrough
How to Find the Vertex of a Parabola
NancyPi — clear walkthrough of the vertex formula

Key Concepts

  • A relation is a function when each input has exactly one output.
  • f(a) means substitute a for every input variable.
  • Domain is the set of allowable inputs; range is the resulting outputs.
  • Exclude denominator zeros; require an even-root radicand to be nonnegative.
\operatorname{dom}\!\left(\frac{1}{x-a}\right)=(-\infty,a)\cup(a,\infty)
⚡ Repeated outputs are allowed. A repeated input with two different outputs is not a function.
No formulas are provided on Test 1. This sheet is a study aid only. You may not use it during the test, so practice writing every formula from memory.

📐 Core Algebra

Difference of Squares
a^2-b^2=(a-b)(a+b)
Difference of Cubes
a^3-b^3=(a-b)(a^2+ab+b^2)
Sum of Cubes
a^3+b^3=(a+b)(a^2-ab+b^2)
Quadratic Formula
x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}

📏 Coordinate Geometry

Midpoint
M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
Distance
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
Slope
m=\frac{y_2-y_1}{x_2-x_1}
Circle
(x-h)^2+(y-k)^2=r^2

📈 Lines

Point-Slope
y-y_1=m(x-x_1)
Slope-Intercept
y=mx+b
Parallel
m_1=m_2
Perpendicular
m_1 \cdot m_2=-1

⛰️ Quadratic Vertex

Vertex x-coordinate
x=-\frac{b}{2a}
Rule
a > 0 → min | a < 0 → max

🔢 Exponent Rules

Product
a^m \cdot a^n=a^{m+n}
Quotient
\frac{a^m}{a^n}=a^{m-n}
Power
(a^m)^n=a^{mn}
Negative
a^{-n}=\frac{1}{a^n}

🔁 Functions & Domain

Function value
f(a):\;\text{substitute }a\text{ for every }x
Rational domain
\text{denominator}\ne0
Even-root domain
\text{radicand}\ge0
Graph test
A vertical line intersects a function graph at most once.
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