Write the Equation of a Linear Function Part II
Learning Outcomes
- Write the equation of a linear function from a graph
- Write the equation for a linear function from slope given two points
Now that we have written equations for linear functions using both slope-intercept form and point-slope form, we can choose which method to use based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the graph of the function f below.
We are not given the slope of the line, but we can choose any two points on the line to find the slope. Choose [latex](0, 7)[/latex] and [latex](4, 4)[/latex]. We can use these points to calculate the slope.
Now we can substitute the slope and the coordinates of one of the points into point-slope form.
If we want to rewrite the equation in slope-intercept form, we would do the following:
If we wanted to find the equation of the line in slope-intercept form without first using point-slope form, we could have recognized that the line crosses the y-axis when the output value is [latex]7[/latex]. Therefore, b = [latex]7[/latex]. We now have the initial value b and the slope m so we can substitute m and b into slope-intercept form of a line.
The function is [latex]f\left(x\right)=-\dfrac{3}{4}x+7[/latex], and the linear equation would be [latex]y=-\dfrac{3}{4}x+7[/latex].
How To: Given the graph of a linear function, write an equation to represent the function
- Identify two points on the line.
- Use the two points to calculate the slope.
- Determine where the line crosses the y-axis to identify the y-intercept by visual inspection.
- Substitute the slope and y-intercept into slope-intercept form of a linear equation.
Example
Write an equation for a linear function f given its graph shown below.Answer:
Identify two points on the line such as [latex](0, 2)[/latex] and [latex](–2, –4)[/latex]. Use the points to calculate the slope.
Substitute the slope and the coordinates of one of the points into point-slope form.
We can use algebra to rewrite the equation in slope-intercept form.
[latex-display]\begin{array}{l}y+4=3\left(x+2\right)\hfill \\ y+4=3x+6\hfill \\ y=3x+2 \hfill \end{array}[/latex-display] This makes sense because we can see from the graph above that the line crosses the y-axis at the point [latex](0,2)[/latex], which is the y-intercept, so [latex]b=2[/latex]Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex 1: Find the Equation of a Line in Slope Intercept Form Given the Graph of a Line. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- College Algebra. Provided by: OpenStax Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download free at : http://cnx.org/contents/[email protected]:1/Preface.
- Ex: Find the Linear Function Given Two Function Values in Function Notation. Authored by: for Lumen Learning. License: CC BY: Attribution.
Example
If f is a linear function, with [latex]f\left(3\right)=-2[/latex] , and [latex]f\left(8\right)=1[/latex], find an equation for the function in slope-intercept form.Answer:
We can write the given points using coordinates.
We can then use the points to calculate the slope.
Substitute the slope and the coordinates of one of the points into point-slope form.
We can use algebra to rewrite the equation in slope-intercept form.
Summary