Functions

Unit Review Sheet

These facts and definitions should be mastered throughout this unit. This page can be used for periodic review and study as you are finishing the unit and in the future.

Facts and Definitions

Lesson 1: What Is a Function?

A function assigns exactly one output for every input.
The Vertical Line Test states that if your vertical line touches a graph at more than one point at the same time, then the graph is not a function.
A linear function is a function that creates a straight line when graphed.
Linear functions change at a constant rate. When you increase the input ( $x$ ), the output ( $y$ ) also increases or decreases consistently.
A non-linear function is a function that is not a straight line.

Lesson 2: Linear and Nonlinear

If an equation includes a power (like $x^{2}$ or $x^{3}$ ) or a root (like $\sqrt{4}$ or $\sqrt[3]{27}$ ), then the relationship will NOT be linear.

Lesson 3: Understanding Functions

Understanding the shape of a graph (whether it is increasing, decreasing, or constant) helps you match it to a real-world situation.

Lesson 4: Intercepts

An intercept is a point where a line crosses either the x-axis or the y-axis on a graph.
The x-intercept is the point where a line crosses the x-axis, and at this point the y-coordinate is always 0.
The y-intercept is the point where a line crosses the y-axis, and at this point the x-coordinate is always 0.
To find the x-intercept from a table, look for the point where $y = 0$ ; if $y = 0$ is not listed, estimate the x-intercept by finding the pattern between the points.
To find the y-intercept from a table, look for the point where $x = 0$ ; if $x = 0$ is not listed, estimate the y-intercept by finding the pattern between the points.
To find the x-intercept from an equation, set $y = 0$ and solve for $x$ . To find the y-intercept from an equation, set $x = 0$ and solve for $y$ .

Lesson 5: Slope

Slope describes how steep a line is and the direction it goes on a graph.
The formula for slope is $\frac{rise}{run}$ . It's also written as $\frac{Δy}{Δx}$ , where the triangle (Δ) means "change in."
Rise is how much the line goes up or down.
Run is how much the line goes left or right.
A positive slope means the line goes up as you move from left to right.
A negative slope means the line goes down as you move from left to right.
Horizontal lines have a zero slope because they don't rise or fall — they stay flat.
Vertical lines have an undefined slope because they don't run left or right — and dividing by zero is not possible.
Use the formula: Slope = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ to find the slope between two points.
A table of values can be used to find slope by treating each row as a point ( $x, y$ ). Use the formula: Slope = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ with any two rows from the table.

Lesson 6: Slope-Intercept Form

The Slope-Intercept Form of a line is $y = m x + b$ where m = slope and b = y-intercept.
The Standard Form of an equation is $A x + B y = C$ .
To graph an equation in standard form, first rewrite it in slope-intercept form ( $y = m x + b$ ).

Lesson 7: Creating Functions

The input is the value you choose or control in a function, often represented by a variable like $x$ .
The output is the value that depends on the input and is calculated using the function rule.
The rate of change tells how much the output increases or decreases each time the input goes up by one.
The rate of change in a linear equation is also the slope.
A linear function rule is the formula that describes how to find the output from the input, usually written in the form: output = slope × input + starting value

Lesson 8: Comparing Functions

The rate of change can be found by dividing the change in quantity by the change in time (e.g., miles per minute).
On a graph, the slope is calculated by identifying two points and applying the slope formula.
A steeper slope indicates a faster rate of change in the function.

Lesson 9: Unit 5 Test

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Lesson 10: Final Project

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