Algebra 3: Linear Functions - Mathematics Platform

Learning Objectives

Understand what a function is and how it relates to equations
Identify linear functions from equations and graphs
Find slope and y-intercept from equations and graphs
Write linear equations in slope-intercept form
Graph linear functions by plotting points
Interpret real-world linear relationships

1. Understanding Functions

What is a Function?

A function is a special relationship between inputs and outputs where each input has exactly one output.

Think of a function as a machine:

Input (x)

→

f(x) = 2x + 1

→

Output (y)

Function Examples:

f(x) = 2x + 3 → f(1) = 5, f(2) = 7, f(3) = 9
g(x) = x² → g(1) = 1, g(2) = 4, g(3) = 9
h(x) = 3x - 1 → h(1) = 2, h(2) = 5, h(3) = 8

Function Machine

Test different inputs with the function: f(x) = 2x + 3

Input x =

f(1) = 5

Function Table:

x	f(x)
-2	-1
-1	1
0	3
1	5
2	7

2. Linear Functions

What are Linear Functions?

Linear functions have graphs that are straight lines. They can be written in the form y = mx + b

Slope-Intercept Form:

y = mx + b

m = slope (steepness of the line)
b = y-intercept (where line crosses y-axis)

Slope & Intercept Explorer

Slope (m): 1

Y-intercept (b): 0

Current equation: y = 1x + 0

3. Finding the Slope

Slope Formula

Slope measures how steep a line is. For two points (x₁, y₁) and (x₂, y₂):

m = \frac{y_2 - y_1}{x_2 - x_1}

Types of Slope:

Positive Slope: Line rises from left to right

Negative Slope: Line falls from left to right

Zero Slope: Horizontal line

Undefined Slope: Vertical line

Slope Calculator

Point 1:

x₁ = y₁ =

Point 2:

x₂ = y₂ =

4. Graphing Linear Functions

How to Graph Linear Functions

Find the y-intercept (b): This is where the line crosses the y-axis
Use the slope (m): From the y-intercept, move up/down and right/left according to the slope
Plot points: Connect the points with a straight line
Check: Choose a point and verify it satisfies the equation

Graphing Practice

Graph the line: y = 2x + 1

Step 1: Find y-intercept

When x = 0: y = 2(0) + 1 = ✓

Step 2: Use slope to find another point

Slope = 2, so rise 2, run 1. From (0,1): go to (, ) ✓

5. Real-World Linear Functions

Real-World Examples

Linear functions appear everywhere in real life!

Business: Cell Phone Plans

A cell phone company charges $20 per month plus $0.10 per minute:

Total cost = 0.10 × minutes + 20

Physics: Distance vs Time

A car traveling at 60 mph:

Distance = 60 × time

Temperature Conversion

Celsius to Fahrenheit: °F = (⅛)°C + 32

Real-World Application

A lemonade stand sells cups for $2 each. The owner has fixed costs of $10 (for supplies).

Profit = 2 × cups sold - 10

1. How much profit if they sell 8 cups?

2. How many cups to break even (profit = 0)?