What are the slope and y-intercept of the linear function graphed to the left?

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Learning Outcome

• Graph a linear function using the slope and y-intercept

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.

A General Note: Graphical Interpretation of a Linear Function

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

• b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis.
• m 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=\dfrac{\text{change in output (rise)}}{\text{change in input (run)}}=\dfrac{\Delta y}{\Delta x}=\dfrac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]

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.)

How To: Given the equation for a linear function, graph the function using the y-intercept and slope

1. Evaluate the function at an input value of zero to find the y-intercept.
2. Identify the slope.
3. Plot the point represented by the y-intercept.
4. Use [latex]\dfrac{\text{rise}}{\text{run}}[/latex] to determine at least two more points on the line.
5. Sketch the line that passes through the points.

Example

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

Try It

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.

Video transcript

We are asked to graph y is equal to 1/3x minus 2. Now, whenever you see an equation in this form, this is called slope-intercept form. And the general way of writing it is y is equal to mx plus b, where m is the slope. And here in this case, m is equal to 1/3-- so let me write that down-- m is equal to 1/3, and b is the y-intercept. So in this case, b is equal to negative 2. And you know that b is the y-intercept, because we know that the y-intercept occurs when x is equal to 0. So if x is equal to 0 in either of these situations, this term just becomes 0 and y will be equal to b. So that's what we mean by b is the y-intercept. So whenever you look at an equation in this form, it's actually fairly straightforward to graph this line. b is the y-intercept. In this case it is negative 2, so that means that this line must intersect the y-axis at y is equal to negative 2, so it's this point right here. Negative 1, negative 2, this is the point 0, negative 2. If you don't believe me, there's nothing magical about this, try evaluating or try solving for y when x is equal to 0. When x is equal to 0, this term cancels out and you're just left with y is equal to negative 2. So that's the y-intercept right there. Now, this 1/3 tells us the slope of the line. How much do we change in y for any change in x? So this tells us that 1/3, so that right there, is the slope. So it tells us that 1/3 is equal to the change in y over the change in x. Or another way to think about it, if x changes by 3, then y would change by 1. So let me graph that. So we know that this point is on the graph, that's the y-intercept. The slope tells us that if x changes by 3-- so let me go 3 three to the right, 1, 2, 3-- that y will change by 1. So this must also be a point on the graph. And we could keep doing that. If x changes by 3, y changes by 1. If x goes down by 3, y will go down by 1. If x goes down by 6, y will go down by 2. It's that same ratio, so 1, 2, 3, 4, 5, 6, 1, 2. And you can see all of these points are on the line, and the line is the graph of this equation up here. So let me graph it. So it'll look something like that. And you're done.

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