Graphs of Linear Functions: Learn It 6

Graphing a Linear Function Using [latex]y[/latex]-intercept and Slope

We can graph linear functions by using specific characteristics of the function rather than plotting points. The first characteristic is its [latex]y[/latex]–intercept which is the point at which the input value is zero. To find the [latex]y[/latex]–intercept, we can set [latex]x=0[/latex] in the equation. The other characteristic of the linear function is its slope [latex]m[/latex].

Remember that if a function has a [latex]y[/latex]-intercept, we can always find it by setting [latex]x=0[/latex] and then solving for [latex]y[/latex].
Let’s consider the following function:

[latex]f\left(x\right)=\frac{1}{2}x+1[/latex]

  • The slope is [latex]\frac{1}{2}[/latex]. Because the slope is positive, we know the graph will slant upward from left to right.
  • The [latex]y[/latex]–intercept is the point on the graph when [latex]x = 0[/latex]. The graph crosses the [latex]y[/latex]-axis at [latex](0, 1)[/latex].

Now we know the slope and the [latex]y[/latex]-intercept. We can begin graphing by plotting the point [latex](0, 1)[/latex] We know that the slope is rise over run, [latex]m=\frac{\text{rise}}{\text{run}}[/latex].

From our example, we have [latex]m=\frac{1}{2}[/latex], which means that the rise is [latex]1[/latex] and the run is [latex]2[/latex]. Starting from our [latex]y[/latex]-intercept [latex](0, 1)[/latex], we can rise [latex]1[/latex] and then run [latex]2[/latex] or run [latex]2[/latex] and then rise [latex]1[/latex]. We repeat until we have multiple points, and then we draw a line through the points as shown below.

graph of the line y = (1/2)x +1 showing the "rise", or change in the y direction as 1 and the "run", or change in x direction as 2, and the y-intercept at (0,1)

graphical interpretation of a linear function

In the equation [latex]f\left(x\right)=mx+b[/latex]

  • [latex]b[/latex] is the [latex]y[/latex]-intercept of the graph and indicates the point [latex](0, b)[/latex] at which the graph crosses the [latex]y[/latex]-axis.
  • [latex]m[/latex] is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:

[latex]m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]

Match each equation of the linear functions with one of the lines on the graph.

  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]

Graph of three lines, line 1) passes through (0,3) and (-2, -1), line 2) passes through (0,3) and (-6,0), line 3) passes through (0,-3) and (2,1)

How To: Given the equation for a linear function, graph the function using the [latex]y[/latex]-intercept and slope.

  1. Evaluate the function at an input value of zero to find the [latex]y[/latex]–intercept.
  2. Identify the slope.
  3. Plot the point represented by the y-intercept.
  4. Use [latex]\frac{\text{rise}}{\text{run}}[/latex] to determine at least two more points on the line.
  5. Draw a line which passes through the points.
Graph [latex]f\left(x\right)=-\frac{2}{3}x+5[/latex] using the [latex]y[/latex]–intercept and slope.

Look at the graph of the function [latex]f[/latex] given below:
This graph shows a linear function graphed on an x y coordinate plane. The x axis is labeled from negative 2 to 8 and the y axis is labeled from negative 1 to 8. The function f is graph along the points (0, 7) and (4, 4).

We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose [latex](0, 7)[/latex] and [latex](4, 4)[/latex].

[latex]\begin{array}{rcl} m & = & \frac{y_2 - y_1}{x_2 - x_1} \\ m & = & \frac{4 - 7}{4 - 0} \\ m & = & -\frac{3}{4} \end{array}[/latex]

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.

[latex]\begin{array}{rcl} y - y_1 & = & m(x - x_1) \\ y - 4 & = & -\frac{3}{4}(x - 4) \end{array}[/latex]

If we want to rewrite the equation in the slope-intercept form, we would find

[latex]\begin{array}{rcl} y - 4 & = & -\frac{3}{4}(x - 4) \\ y - 4 & = & -\frac{3}{4}x + 3 \\ y & = & -\frac{3}{4}x + 7 \end{array}[/latex]

If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7. Therefore, [latex]b = 7[/latex]. We now have the initial value [latex]b[/latex] and the slope [latex]m[/latex], so we can substitute [latex]m[/latex] and [latex]b[/latex] into the slope-intercept form of a line.

This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.

So the function is [latex]f(x) = -\frac{3}{4}x + 7[/latex], and the linear equation would be [latex]y = -\frac{3}{4}x + 7[/latex].

How to: Given the graph of a linear function, write an equation to represent the function.

  1. Identify two points on the line.
  2. Use the two points to calculate the slope.
  3. Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.
  4. Substitute the slope and y-intercept into the slope-intercept form of a line equation.