1. Home
  2. College algebra
  3. Linear functions
  4. Linear functions

4.1 Linear functions  (Page 9/27)

<< Chapter < Page
  College algebra     Page 9 / 27
Chapter >> Page >

Vertical stretch or compression

In the equation   f ( x ) = m x , the   m   is acting as the vertical stretch    or compression of the identity function. When   m   is negative, there is also a vertical reflection of the graph. Notice in [link] that multiplying the equation of   f ( x ) = x   by   m   stretches the graph of   f   by a factor of   m   units if   m > 1   and compresses the graph of   f   by a factor of   m   units if   0 < m < 1.   This means the larger the absolute value of   m ,   the steeper the slope.

This graph shows seven versions of the function, f of x = x on an x, y coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. Seven multi-colored lines run through the point (0, 0). Starting with the lines in the top right quadrant and moving clockwise, the first line is f of x = 3 times x and has a slope of 3, the next line is f of x = 2 times x which has a slope of 2, the next line is f of x = x which has a slope of 1, the next line is f of x = x divided by 2 which has a slope of .5. The last line in this quadrant is f of x = x divided by 3 which has a slope of one third x. In the bottom right quadrant moving clockwise, the first line is f of x = negative x divided by 2, which has a slope of negative one half, the middle line is f of x = negative x which has a slope of negative 1, and the last line is f of x = negative 2 times x which has a slope of negative 2.
Vertical stretches and compressions and reflections on the function   f ( x ) = x

Vertical shift

In   f ( x ) = m x + b , the   b   acts as the vertical shift    , moving the graph up and down without affecting the slope of the line. Notice in [link] that adding a value of   b   to the equation of   f ( x ) = x   shifts the graph of   f   a total of   b   units up if   b   is positive and   | b |   units down if   b   is negative.

This graph shows six versions of the function, f of x = x, on an x, y coordinate plane. The x-axis runs from negative 8 to 8, and the y axis runs negative 8 to 8. There are five lines parallel to each other. The first line extends from the bottom left quadrant to the upper right quadrant on the coordinate plane. This line shows f of x = x plus 4 which has a slope of 1 and a y-intercept at 4. The next line also extends from the bottom left quadrant to the upper right quadrant and shows f of x = x plus 2 which has a slope of 1 and a y-intercept at 2. The next and middle line, extends from the lower left quadrant, through the center of the graph at point (0, 0) to the upper right quadrant and shows f of x = x. The next line extends from the lower left quadrant, through the lower right quadrant to the upper right quadrant. This line shows f of x = x minus 2 which has a slope of 1 and a y-intercept at -2. The last line extends from the lower left quadrant, through the lower right quadrant to the upper right quadrant.This line shows f of x = x minus 4 which has a slope of 1 and a y-intercept at -4.
This graph illustrates vertical shifts of the function f ( x ) = x .

Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice each method.

Given the equation of a linear function, use transformations to graph the linear function in the form   f ( x ) = m x + b .

  1. Graph   f ( x ) = x .
  2. Vertically stretch or compress the graph by a factor   m .
  3. Shift the graph up or down   b   units.

Graphing by using transformations

Graph   f ( x ) = 1 2 x − 3   using transformations.

The equation for the function shows that   m = 1 2   so the identity function is vertically compressed by   1 2 .   The equation for the function also shows that   b = − 3   so the identity function is vertically shifted down 3 units. First, graph the identity function, and show the vertical compression as in [link] .

This graph shows two functions on an x, y coordinate plane. One shows an increasing function of y = x divided by 2 that runs through the points (0, 0) and (2, 1). The second shows an increasing function of y = x and runs through the points (0, 0) and (1, 1)).
The function,   y = x , compressed by a factor of   1 2

Then show the vertical shift as in [link] .

This graph shows two functions on an x, y coordinate plane. The first is an increasing function of y = x divided by 2 and runs through the points (0, 0) and (2, 1). The second shows an increasing function of y = x divided by 2 minus 3 and passes through the points (0, 3) and (2, -2). An arrow pointing downward from the first function to the second function reveals the vertical shift.
The function   y = 1 2 x , shifted down 3 units
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph   f ( x ) = 4 + 2 x   using transformations.

This graph shows three functions on an x, y coordinate plane. One shows an increasing function y = x that passes through points (0, 0) and (2, 2). A second shows an increasing function y = 2 times x that passes through the points (0, 0) and (2, 4). The third is an increasing function y = 2 times x plus 4 and passes through the points (0, 4) and (2, 8).
Got questions? Get instant answers now!

In [link] , could we have sketched the graph by reversing the order of the transformations?

No. The order of the transformations follows the order of operations. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. This is why we performed the compression first. For example, following the order: Let the input be 2.

f ( 2 ) = 1 2 ( 2 ) − 3 = 1 − 3 = −2

Writing the equation for a function from the graph of a line

Earlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing linear functions to analyze graphs a little more closely. Begin by taking a look at [link] . We can see right away that the graph crosses the y -axis at the point   ( 0 , 4 )   so this is the y -intercept.

This graph shows the function f of x = 2 times x plus 4 on an x, y coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. This function passes through the points (-2, 0) and (0, 4).

Then we can calculate the slope by finding the rise and run. We can choose any two points, but let’s look at the point   ( – 2 , 0 ) .   To get from this point to the y- intercept, we must move up 4 units (rise) and to the right 2 units (run). So the slope must be

m = rise run = 4 2 = 2

Substituting the slope and y- intercept into the slope-intercept form of a line gives

Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²
Mamush Reply
how can I find set theory
Ephraim Reply
how can I find set theory
Jarvis
is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m
Patrick Reply
hi, interested in algebra
Makan Reply
how to reduce an equation?
Makan
by manipulation of both side
Al
9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why
Patrick Reply
when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.
Philip
Given a polynomial expression, factor out the greatest common factor.
Hanu Reply
WHAT IS QUADRATIC EQUATION?
Charles Reply
WHAT IS SYSTEM OF LINEAR INEWUALITIES?
Charles
WHAT IS SYSTEM OF LINEAR INEWUALITIES?
Charles
complex perform
Angel
what is equation?
Charles Reply
what are equations?
Charles
Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K
Rakiya
Please help me is assignment
Rakiya
The 47th problem of Euclid
Kenneth
show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 10

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask