Another way to graph a linear function is by using its slope m and y-intercept.

Let us consider the following function.

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

The function is in slope-intercept form, so the slope is [latex]\dfrac{1}{2}[/latex]. Because the slope is positive, we know the graph will slant upward from left to right. The y-intercept is the point on the graph when [latex]x=0[/latex]. The graph crosses the y-axis at [latex](0, 1)[/latex]. Now we know the slope and the y-intercept. We can begin graphing by plotting the point [latex](0, 1)[/latex]. We know that the slope is rise over run, [latex]m=\dfrac{\text{rise}}{\text{run}}[/latex]. From our example, we have [latex]m=\dfrac{1}{2}[/latex], which means that the rise is [latex]1[/latex] and the run is [latex]2[/latex]. So starting from our y-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 a few points and then we draw a line through the points as shown in the graph below.

All linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line parallel to the y-axis does not have a y-intercept, but it is not a function.)

Example

Graph [latex]f\left(x\right)=-\dfrac{2}{3}x+5[/latex] using the y-intercept and slope.

Show Solution

Evaluate the function at [latex]x=0[/latex] to find the y-intercept. The output value when [latex]x=0[/latex] is [latex]5[/latex], so the graph will cross the y-axis at [latex](0, 5)[/latex].

According to the equation for the function, the slope of the line is [latex]-\dfrac{2}{3}[/latex]. This tells us that for each vertical decrease in the “rise” of [latex]–2[/latex] units, the “run” increases by [latex]3[/latex] units in the horizontal direction. We can now graph the function by first plotting the y-intercept in the graph below. From the initial value [latex](0, 5)[/latex], we move down [latex]2[/latex] units and to the right [latex]3[/latex] units. We can extend the line to the left and right by using this relationship to plot additional points and then drawing a line through the points.

The graph slants downward from left to right, which means it has a negative slope as expected.

In the following video we show another example of how to graph a linear function given the y-intercepts and the slope.

In the last example, we will show how to graph another linear function using the slope and y-intercept.

Example

Graph [latex]f\left(x\right)=-\dfrac{3}{4}x+6[/latex] using the slope and y-intercept.

Show Solution

The slope of this function is [latex]-\dfrac{3}{4}[/latex] and the y-intercept is [latex](0,6)[/latex] We can start graphing by plotting the y-intercept and counting down three units and right [latex]4[/latex] units. The first stop would be [latex](4,3)[/latex], and the next stop would be [latex](8,0)[/latex].