Algebra 1 EOC Test pt. 2

Whole Numbers

0, 1, 2, …

Integers

-3, -2, -1, 0, 1, 2, 3, …

Rational Number

Any number that can be written as a fraction

(Includes proper/improper fractions, mixed numbers, terminating decimals, repeating decimals with a pattern, integers, and whole numbers)

Irrational Number

Any number that cannot be written as a fraction

(Includes only repeating decimals with no pattern)

Opposites

A and -A

Ex: -5 and 5

Absolute Value

Distance between a number and zero (always positive)

Functions

Each input is paired with exactly one output (no repeating input values)

Zero of a Function

X-intercept (find x-value when y or f(x) equals zero)

Domain

Input or X-values

Range

Output or Y-values

Vertical Line Test

Used to determine whether a graphed relation is a function

1) Draw a vertical line through every point on the graph

2) If each vertical line drawn intersects only one point, i is a function

3) If any vertical line drawn intersects more than one point, it is not a function

Slope-Intercept Form

y=mx + b

m=slope

b=y-intercept

Standard Form

Ax+Bx=C

A, B, & C must be integers

Point-Slope Form

y-y₁=m(x-x₁)

m=slope

(x₁,y₁)=point on the line

X-Intercept

Where the graph crosses the x axis (x,0)

Set y=0 to find

Y-Intercept

Where the graph crosses the y axis (0,y)

Set x=0 to find

Horizontal Line

y=b

All points have the same y value

Vertical Line

x=a

All points have the same x value

Parallel Lines

Two lines with the same slope

Perpendicular Lines

Two lines with opposite reciprocal slopes

Ex: 3 and -1/3

Slope Formula

Δy/Δx=y₂-y₁/x₂-x₁

Direct Variation

y=kx where k= constant of variation (slope of line) k=y/x

The graph is a line that always passes through the origin

Solving an Inequality

Reverse the inequality sign if you multiply or divide both sides of the inequality by a negative number

Graphing Inequalities on a Number Line

x>a or x<a = Open circle at a

Shade in the direction of your solution

x≥a or x≤a = closed circle at a

Shade in the direction of your solution

Graphing a Linear Inequality

1) Graph the boundary line (≥ or ≤ = solid line, > or < = dashed line)

2) Pick a test point to determine which half of the graph to shade

If solving a system of linear ineqalities, the solution is where the shaded regions overlap

Absolute Value Equations

|x|=4

x=4 or x=-4

|x|=6

x= no solution

Absolute Value Inequalities

Less Than= forms "and" compound inequality (Less Th-AND)

|x|<8

-8<x<8

Greater Than= forms "or" compound inequality (Great-OR Than)

|x|>10

x

Methods of Solving a Linear System

1) Graphing (solution= where the two lines intersect)

2) Substitution (use if one equation is solved for x or y)

3) Elimination (use if both equations are in standard form

Special Cases

If both variables cancel out, the answer is either no solution (false statement) or many solutions (true statement)

Arithmetic Sequence

a_n=a₁ + (n-1)d

n= nth term

a₁=1st term

d= common difference (+ or - pattern)

Geometric Sequence

a_n=a₁ × r^n-¹

Exponent Rules

a^m × a^n=a^m+n Ex: 4³ × 4⁶=4⁹

a^m/a^n=a^m-n Ex: 3¹⁰/3⁴=3⁶

(a^m)^n=a^mn Ex: (6²)⁵=6¹⁰

(ab)^m=a^m × b^m Ex: (2×5)³=2³×5³

(a/b)^m=a^m/b^m Ex:(¾)²=3²/4²

a⁰=1 Ex: 8⁰=1

a^-n= 1/a^n Ex: 5^-3=¹/₅³

a^m/n=n√a^m Ex: 25³/²= (√25)³

Foil

Multiply first, outer, inner, and last terms

Sum and Difference (Special Product)

(a+b)(a-b)=a²-b²

Squaring a Binomial (Special Product)

(a+/_b)²=a²+/-2ab+b²

Quadratic Function

y=ax²+bx+c

Graph is a parabola or u shaped

If a is positive, parabola faces up

If a is negative, parabola faces down

Use x=-b/2a to find the axis of symmetry and the x coordinate of the vertex

Zeros of a Quadratic Function

x-intercepts on the parabola

Methods for Solving a Quadratic Equation

1)Factoring:Equation must be set equal to zero before factored

2)Square roots: only used if b-term is missing (ax²+c=0)

3) Completing the square: Equation must be in the form ax²+bx=c; you need to factor out the leading coefficient if a≉1

4)Quadratic formula: USed if the equation cannot be factored; you must set the equation equal to zero first

X=-b+/-√b²-4ac/2a

Discriminant

Used to determine the number of solutions for a quadratic equation

Positive: 2 solutions

Zero: 1 solution

Negative: no solutions

b²-4ac

Exponential Functions

Graph forms a smooth curve that approaches the x axis but never intersects it

y=a×b^x where a ≉ 0, b≉1, and b>0