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Study Guides > Intermediate Algebra

Graph Linear Functions Using Slope and y-Intercept

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. graph of the line y = (1/2)x +1 showing the

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.

Answer: 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. graph of the line y = (-2/3)x + 5 showing the change of -2 in y and change of 3 in x. The graph slants downward from left to right, which means it has a negative slope as expected.

Try It

[ohm_question]79774[/ohm_question]
In the following video we show another example of how to graph a linear function given the y-intercepts and the slope. https://youtu.be/N6lEPh11gk8 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.

Answer: 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]. Screen Shot 2016-07-13 at 1.35.32 PM

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • Ex: Graph a Line and ID the Slope and Intercepts (Fraction Slope). Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
  • QID 79774: Graph linear eq. in slope-intercept form, give slope and intercept.. Authored by: Day, Alyson. License: CC BY: Attribution.

CC licensed content, Specific attribution

  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..