CheatSheetsHero

brayden-anderson / Algbra 1 final

June 17, 2025 02:46
algebra
equations
functions
quadratics
statistics
Download PDF
Missing something?

Algbra 1 final

A comprehensive cheat sheet covering essential Algebra I concepts, formulas, and techniques. This guide includes data representation, linear equations, functions, exponential functions, and quadratics.

Data Representation and Analysis

Data Representations

Dot Plot:
Simple way to represent data, each dot represents a single observation.

Histogram:
Groups data into bins and displays the frequency of each bin as a bar. No spaces between bars.

Box and Whisker Plot:
Displays the five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

Useful for identifying the spread and skewness of the data.

Measures of Central Tendency and Spread

Mean:
Average of all data points. Sum of values divided by the number of values.
Mean = (sum of x) / n

Median:
Middle value when data is ordered. If there are two middle values, average them.

Mode:
Value that appears most frequently in the data set.

Range:
Difference between the maximum and minimum values.
Range = Max - Min

Standard Deviation:
Measures the spread of data around the mean. A lower standard deviation indicates data points are closer to the mean.

IQR (Interquartile Range):
Difference between the third quartile (Q3) and the first quartile (Q1).
IQR = Q3 - Q1

Outliers

Outliers:
Data points that are significantly different from other data points in the set.
Can be identified using the 1.5 * IQR rule:
Lower Bound = Q1 - 1.5 * IQR
Upper Bound = Q3 + 1.5 * IQR
Values outside these bounds are considered outliers.

Linear Equations, Inequalities, and Systems

Linear Equations

Standard Form:
Ax + By = C

Slope-Intercept Form:
y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form:
y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Solving Linear Equations:
Use inverse operations to isolate the variable. Simplify both sides before isolating the variable.

Systems of Equations

Substitution Method:
Solve one equation for one variable and substitute that expression into the other equation.

Elimination Method:
Add or subtract multiples of the equations to eliminate one variable.

Applications:
Mixture problems, wind/current problems, age problems, coin problems, perimeter problems.

Solving Systems:
Find the values of the variables that satisfy all equations in the system. Represented as an ordered pair (x, y).

Linear Inequalities

Graphing Linear Inequalities:
Graph the boundary line (dashed for < or >, solid for ≤ or ≥). Shade the region that satisfies the inequality.

Systems of Inequalities:
Graph each inequality and find the overlapping shaded region, which represents the solution set.

Linear Programming:
Optimize an objective function subject to constraints. Graph the constraints and find the feasible region. The optimal solution occurs at a vertex of the feasible region.

Example:
y > 2x + 1

Functions and Function Notation

Function Basics

Function Notation:
f(x) represents the value of the function f at x.

Writing Functions:
Express the relationship between input (x) and output (f(x)).

Tables:
Represent function values in a table format with input and output values.

Graphing:
Plot points (x, f(x)) on a coordinate plane to visualize the function.

Types of Functions

Absolute Value Function:
f(x) = |x|
V-shaped graph with vertex at (0,0).

Piecewise Functions:
Defined by different expressions over different intervals of the domain.

Example:

f(x) = { x^2, x < 0
        { x + 1, x >= 0

Absolute Value as Piecewise:
|x| = { x, x >= 0 {-x, x < 0

Example:
f(x) = |x-2|

Domain and Range

Domain:
Set of all possible input values (x) for which the function is defined.

Range:
Set of all possible output values (f(x)) of the function.

Set Builder Notation:
{x | condition}
Example: {x | x > 0}

Interval Notation:
Use brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints.
Example: (0, ∞)

Inverse Functions

Inverse Function:
If f(x) maps x to y, then f⁻¹(y) maps y back to x.
To find the inverse, swap x and y and solve for y.

Exponential Functions

Exponential Function Basics

General Form:
f(x) = a * b^x
where a is the initial value and b is the base (growth/decay factor).

Exponential Growth:
b > 1
The function increases as x increases.

Exponential Decay:
0 < b < 1
The function decreases as x increases.

Solving Exponential Equations:
Use logarithms or rewrite with a common base.

Exponent Rules

Exponent Rules Summary:

  • x^m * x^n = x^(m+n)
  • (x^m) / (x^n) = x^(m-n)
  • (x^m)^n = x^(m*n)
  • (xy)^n = x^n * y^n
  • (x/y)^n = x^n / y^n
  • x^0 = 1
  • x^(-n) = 1 / x^n

Compound Interest

Compound Interest Formula:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for

Percent of Change

Percent of Change Formula:
Percent Change = ((New Value - Old Value) / Old Value) * 100

  • Positive result indicates percent increase.
  • Negative result indicates percent decrease.

Quadratic Functions

Factoring

Factoring Quadratics:
Expressing a quadratic expression as a product of two binomials.
Example: x^2 + 5x + 6 = (x + 2)(x + 3)

Zero Product Property:
If ab = 0, then a = 0 or b = 0.
Used to solve factored quadratic equations.

Difference of Squares:
a^2 - b^2 = (a + b)(a - b)

Perfect Square Trinomial:
a^2 + 2ab + b^2 = (a + b)^2
a^2 - 2ab + b^2 = (a - b)^2

Quadratic Forms and Graphs

Standard Form:
f(x) = ax^2 + bx + c

Factored Form:
f(x) = a(x - r1)(x - r2), where r1 and r2 are the roots (x-intercepts).

Vertex Form:
f(x) = a(x - h)^2 + k, where (h, k) is the vertex.

Graphing Quadratics:
Parabola shape.
Vertex is the minimum (if a > 0) or maximum (if a < 0) point.

Solving Quadratic Equations

Quadratic Formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Used to find the roots of a quadratic equation in standard form.

Completing the Square:
Transform the quadratic equation into vertex form and solve for x.

Discriminant:
Δ = b^2 - 4ac

  • If Δ > 0, two real solutions.
  • If Δ = 0, one real solution.
  • If Δ < 0, no real solutions (two complex solutions).

Geometric Patterns:
Quadratic functions can represent areas and patterns that grow quadratically.