We previously wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at the graph below. We can see right away that the graph crosses the y-axis at the point (0, 4), so this is the y-intercept.

Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point (–2, 0). To get from this point to the y-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be:

[latex]m=\frac{\text{rise}}{\text{run}}=\frac{4}{2}=2[/latex]

Substituting the slope and y-intercept into slope-intercept form of a line gives:

[latex]y=2x+4[/latex]

Example: Matching Linear Functions to Their Graphs

Match each equation of a linear function with one of the lines in the graph below.

1. [latex]f\left(x\right)=2x+3[/latex]
2. [latex]g\left(x\right)=2x - 3[/latex]
3. [latex]h\left(x\right)=-2x+3[/latex]
4. [latex]j\left(x\right)=\frac{1}{2}x+3[/latex]

Show Solution

Analyze the information for each function.

1. This function has a slope of 2 and a y-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function g has the same slope, but a different y-intercept. Lines I and III have the same slant because they have the same slope. Line III does not pass through (0, 3) so f must be represented by line I.
2. This function also has a slope of 2, but a y-intercept of –3. It must pass through the point (0, –3) and slant upward from left to right. It must be represented by line III.
3. This function has a slope of –2 and a y-intercept of 3. This is the only function listed with a negative slope, so it must be represented by line IV because it slants downward from left to right.
4. This function has a slope of [latex]\frac{1}{2}[/latex] and a y-intercept of 3. It must pass through the point (0, 3) and slant upward from left to right. Lines I and II pass through (0, 3), but the slope of j is less than the slope of f so the line for j must be flatter. This function is represented by Line II.

Now we can re-label the lines.

Finding the x-intercept of a Line

So far we have been finding the y-intercepts of functions: the point at which the graph of a function crosses the y-axis. A function may also have an x-intercept, which is the x-coordinate of the point where the graph of a function crosses the x-axis. In other words, it is the input value when the output value is zero.

To find the x-intercept, set the function f(x) equal to zero and solve for the value of x. For example, consider the function shown:

[latex]f\left(x\right)=3x - 6[/latex]

Set the function equal to 0 and solve for x.

[latex]\begin{array}{l}0=3x - 6\hfill \\ 6=3x\hfill \\ 2=x\hfill \\ x=2\hfill \end{array}[/latex]

The graph of the function crosses the x-axis at the point (2, 0).

Q & A

Do all linear functions have x-intercepts?

No. However, linear functions of the form y = c, where c is a nonzero real number are the only examples of linear functions with no x-intercept. For example, y = 5 is a horizontal line 5 units above the x-axis. This function has no x-intercepts.

Example: Finding an x-intercept

Find the x-intercept of [latex]f\left(x\right)=\frac{1}{2}x - 3[/latex].

Show Solution

Set the function equal to zero to solve for x.

[latex]\begin{array}{l}0=\frac{1}{2}x - 3\\ 3=\frac{1}{2}x\\ 6=x\\ x=6\end{array}[/latex]

The graph crosses the x-axis at the point (6, 0).

Analysis of the Solution

A graph of the function is shown below. We can see that the x-intercept is (6, 0) as expected.

The graph of the linear function [latex]f\left(x\right)=\frac{1}{2}x – 3[/latex].

 

Describing Horizontal and Vertical Lines

There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output or y-value. In the graph below, we see that the output has a value of 2 for every input value. The change in outputs between any two points is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use m = 0 in the equation [latex]f\left(x\right)=mx+b[/latex], the equation simplifies to [latex]f\left(x\right)=b[/latex]. In other words, the value of the function is a constant. This graph represents the function [latex]f\left(x\right)=2[/latex].

A horizontal line representing the function [latex]f\left(x\right)=2[/latex].

A vertical line indicates a constant input or x-value. We can see that the input value for every point on the line is 2, but the output value varies. Because this input value is mapped to more than one output value, a vertical line does not represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula, the denominator will be zero, so the slope of a vertical line is undefined.

Notice that a vertical line has an x-intercept but no y-intercept unless it’s the line x = 0. This graph represents the line x = 2.

The vertical line [latex]x=2[/latex] which does not represent a function.

Example: Writing the Equation of a Horizontal Line

Write the equation of the line graphed below.

Show Solution

For any x-value, the y-value is [latex]–4[/latex], so the equation is [latex]y=–4[/latex].

Example: Writing the Equation of a Vertical Line

Write the equation of the line graphed below.

Show Solution

The constant x-value is 7, so the equation is [latex]x=7[/latex].