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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 .

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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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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