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2.1 Linear functions  (Page 6/17)

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Write the point-slope form of an equation of a line that passes through the points ( –1 , 3 )   and ( 0 , 0 ) . Then rewrite it in the slope-intercept form.

y − 0 = − 3 ( x − 0 ) ; y = − 3 x

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Writing and interpreting an equation for a linear function

Now that we have written equations for linear functions in both the slope-intercept form and the 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 in [link] .

Graph depicting how to calculate the slope of a line

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 ( 0 ,   7 ) and ( 4 ,   4 ) . We can use these points to calculate the slope.

m = y 2 − y 1 x 2 − x 1    = 4 − 7 4 − 0    = − 3 4

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

y − y 1 = m ( x − x 1 )   y − 4 = − 3 4 ( x − 4 )

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

y − 4 = − 3 4 ( x − 4 ) y − 4 = − 3 4 x + 3        y = − 3 4 x + 7

If we wanted 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, b = 7. We now have the initial value b and the slope m so we can substitute m and b into the slope-intercept form of a line.

So the function is f ( x ) = − 3 4 x + 7 , and the linear equation would be y = − 3 4 x + 7.

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.

Writing an equation for a linear function

Write an equation for a linear function given a graph of f shown in [link] .

Graph of an increasing function with points at (-3, 0) and (0, 1).

Identify two points on the line, such as ( 0 ,   2 ) and ( − 2 , −4 ) . Use the points to calculate the slope.

m = y 2 − y 1 x 2 − x 1    = − 4 − 2 − 2 − 0    = − 6 − 2    = 3

Substitute the slope and the coordinates of one of the points into the point-slope form.

y − y 1 = m ( x − x 1 ) y − ( − 4 ) = 3 ( x − ( − 2 ) ) y + 4 = 3 ( x + 2 )

We can use algebra to rewrite the equation in the slope-intercept form.

y + 4 = 3 ( x + 2 ) y + 4 = 3 x + 6        y = 3 x + 2
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Writing an equation for a linear cost function

Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function C where C ( x ) is the cost for x items produced in a given month.

The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is $37.50 for Ben. The variable cost, called the marginal cost, is represented by 37.5. The cost Ben incurs is the sum of these two costs, represented by C ( x ) = 1250 + 37.5 x .

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Writing an equation for a linear function given two points

If f is a linear function, with f ( 3 ) = −2 , and f ( 8 ) = 1 , find an equation for the function in slope-intercept form.

We can write the given points using coordinates.

f ( 3 ) = −2 → ( 3 , −2 ) f ( 8 ) = 1 → ( 8 , 1 )

We can then use the points to calculate the slope.

m = y 2 − y 1 x 2 − x 1    = 1 − ( − 2 ) 8 − 3    = 3 5

Substitute the slope and the coordinates of one of the points into the point-slope form.

     y − y 1 = m ( x − x 1 ) y − ( − 2 ) = 3 5 ( x − 3 )

We can use algebra to rewrite the equation in the slope-intercept form.

y + 2 = 3 5 ( x − 3 ) y + 2 = 3 5 x − 9 5        y = 3 5 x − 19 5
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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