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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 .

Questions & Answers

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
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Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
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rotation by 80 of (x^2/9)-(y^2/16)=1
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what is the standard form if the focus is at (0,2) ?
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a²=4
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A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center
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Practice Key Terms 7

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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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