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How to Graph and Solve a Linear Function Step by Step

Understanding how to graph linear functions is a key Algebra 1 skill. In this guide, we'll explain what linear functions are, how to graph them, and how slope and intercepts work—complete with examples.

What Is a Linear Function?

A linear function is a function whose graph is a straight line. Its general form is:

𝑦 = 𝑚𝑥 + 𝑏

where:

• 𝑚 is the slope (rate of change)
• 𝑏 is the y-intercept (where the line crosses the y-axis)

Key Features of a Linear Function Graph

1. Slope

The slope tells us how steep the line is:

[ \text{Slope} = \frac{\Delta y}{\Delta x} ]

• If a line is horizontal, the slope is 0
• If a line is vertical, the slope is undefined

2. Increasing or Decreasing

• Increasing if 𝑚 > 0 (line goes up)
• Constant if 𝑚 = 0 (line is flat)
• Decreasing if 𝑚 < 0 (line goes down)

3. X-Intercept

The x-intercept is where the line crosses the x-axis (when 𝑦 = 0).

Example: in 𝑦 = 2𝑥 + 3, set 𝑦 = 0:

0 = 2x + 3 → x = −3/2

4. Y-Intercept

The y-intercept is the constant 𝑏 in 𝑦 = 𝑚𝑥 + 𝑏. Set 𝑥 = 0 to find it.

Example: in 𝑦 = 2𝑥 + 3, the y-intercept is (0, 3).

Example Problems

Example 1

Problem: A line passes through (2, 7) and (8, 4). What is the slope?

Solution:

[ \text{Slope} = \frac{4 - 7}{8 - 2} = \frac{-3}{6} = -\frac{1}{2} ]

Example 2

Problem: What are the x-intercept and y-intercept of 𝑦 = 4𝑥 – 5?

Solution:

• y-intercept: set 𝑥 = 0 → 𝑦 = −5, so (0, −5)
• x-intercept: set 𝑦 = 0 → 4𝑥 = 5 → 𝑥 = 5/4, so (5/4, 0)

Summary

• A linear function has the form 𝑦 = 𝑚𝑥 + 𝑏
• Slope (m) tells us the steepness and direction
• Y-intercept (b) is where the line crosses the y-axis (set 𝑥 = 0)
• X-intercept is where the line crosses the x-axis (set 𝑦 = 0)

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