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2.1 Linear functions  (Page 4/17)

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Are the units for slope always units for the output units for the input ?

Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.

Calculate slope

The slope, or rate of change, of a function m can be calculated according to the following:

m = change in output (rise) change in input (run) = Δ y Δ x = y 2 − y 1 x 2 − x 1

where x 1 and x 2 are input values, y 1 and y 2 are output values.

Given two points from a linear function, calculate and interpret the slope.

  1. Determine the units for output and input values.
  2. Calculate the change of output values and change of input values.
  3. Interpret the slope as the change in output values per unit of the input value.

Finding the slope of a linear function

If f ( x ) is a linear function, and ( 3,−2 ) and ( 8,1 ) are points on the line, find the slope. Is this function increasing or decreasing?

The coordinate pairs are ( 3,−2 ) and ( 8,1 ) . To find the rate of change, we divide the change in output by the change in input.

m = change in output change in input = 1 − ( − 2 ) 8 − 3 = 3 5

We could also write the slope as m = 0.6. The function is increasing because m > 0.

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If f ( x ) is a linear function, and ( 2 ,   3 ) and ( 0 ,   4 ) are points on the line, find the slope. Is this function increasing or decreasing?

m = 4 − 3 0 − 2 = 1 − 2 = − 1 2 ; decreasing because m < 0.

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Finding the population change from a linear function

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The rate of change relates the change in population to the change in time. The population increased by 27 , 800 − 23 , 400 = 4400 people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.

4,400  people 4  years = 1,100   people year

So the population increased by 1,100 people per year.

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The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

m = 1 , 868 − 1 , 442 2 , 012 − 2 , 009 = 426 3 = 142  people per year

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Writing the point-slope form of a linear equation

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form    .

y − y 1 = m ( x − x 1 )

The point-slope form is derived from the slope formula.

m = y − y 1 x − x 1 assuming  x ≠ x 1 m ( x − x 1 ) = y − y 1 x − x 1 ( x − x 1 ) Multiply both sides by  ( x − x 1 ) . m ( x − x 1 ) = y − y 1 Simplify . y − y 1 = m ( x − x 1 ) Rearrange .

Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form, y − 4 = − 1 2 ( x − 6 ) . We can convert it to the slope-intercept form as shown.

y − 4 = − 1 2 ( x − 6 ) y − 4 = − 1 2 x + 3 Distribute the  − 1 2 .        y = − 1 2 x + 7 Add 4 to each side .

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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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