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 and free worksheets.

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)

Example:

Here is the graph of the linear function 𝑦 = 2𝑥 + 3:

Key Features of a Linear Function Graph

1. Slope

The slope tells us how steep the line is.

The value of the slope is found by dividing the change in 𝑦 by the change in 𝑥, which is often written as:

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

In 𝑦 = 2𝑥 + 3, the slope 𝑚 = 2.

This means: every time 𝑥 increases by 1, 𝑦 increases by 2.

Special Cases:

• If a line is horizontal, the slope is 0 (e.g., 𝑦 = 5).
• If a line is vertical, the slope is undefined (e.g., 𝑥 = 4).

2. Increasing or Decreasing

The sign of the slope shows whether the line is increasing or decreasing.

A linear function is:

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

Our example 𝑦 = 2𝑥 + 3 is increasing because the slope is positive. On the other hand, a function like 𝑦 = -2𝑥 + 3 has a negative slope, which means it is decreasing.

3. X-Intercept

The x-intercept is where the line crosses the x-axis (when 𝑦 = 0). The x-intercept is also called the zero or root of the linear equation. It represents the value of 𝑥 that makes the linear function equal to zero.

Example: in 𝑦 = 2𝑥+3, set 𝑦 = 0:

\[ 0 = 2x + 3 \quad \Rightarrow \quad x = -\frac{3}{2} \]

The x-intercept is at \( \left(-\frac{3}{2}, 0\right) \).

4. y-Intercept

The y-intercept is where the line crosses the y-axis (when 𝑥 = 0), which is the same value of the constant, 𝑏, in the linear function: 𝑦 = 𝑚𝑥 + 𝑏.

Example: in 𝑦 = 2𝑥 + 3, set 𝑥 = 0:

𝑦 = 2 × 0 + 3 = 3

The y-intercept is at (0, 3).

Example Problems (with Solutions): The Graph of Linear Functions

Example 1

Problem:

The graph of a line passes through the points (2, 7) and (8, 4). What is the slope of the line?

Solution:

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

The slope of the line is

\[ m = -\frac{1}{2} \]

Example 2

Problem:

What are the x-intercept and the y-intercept of the linear function 𝑦 = 4𝑥 – 5?

Solution:

Set 𝑥 = 0, and solve for 𝑦:

𝑦 = 4 × 0 – 5 = -5

So, the y-intercept is (0, -5).

Set 𝑦 = 0, and solve for 𝑥:

0 = 4𝑥 – 5

4𝑥 = 5

𝑥 = \(\frac{5}{4}\)

So, the x-intercept is \( \left(\frac{5}{4}, 0\right) \).

Therefore, the x-intercept is \( \left(\frac{5}{4}, 0\right) \), and the y-intercept is (0, -5).

Summary: Keys to the Graph of a Linear Function

• A linear function is a function whose graph is a straight line. One of its most common forms is 𝑦 = 𝑚𝑥 + 𝑏.
• Slope (m) tells us how steep the line is.
  • A positive slope means the line is increasing (goes up).
  • A negative slope means the line is decreasing (goes down).
  • Slope is calculated as:

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

• The y-intercept (𝑏) is the point where the line crosses the y-axis. To find it, set 𝑥 = 0.
• The x-intercept is where the line crosses the x-axis. To find it, set 𝑦 = 0 and solve for 𝑥.

Additional Math Topics for Algebra 1 – with Free Worksheets

• How to Represent Linear Functions: Slope-Intercept vs Point-Slope Form
• Different Forms of Linear Functions: Standard, Slope-Intercept & Point-Slope
• Linear Functions: The Features of Parallel and Perpendicular

Want more printable practice?

To access more printable worksheets on other math topics, visit this page.

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