Unit 3

Transformations

Read

Part 3.2D corresponds to section 9.1, starting on page 430 of your Pre-Calculus 12 textbook.

In Lesson 1.3, you learned about rational functions. In a rational function, such as y=1x, a variable will be in the denominator. There will be subsequent restrictions on the variable because the value of the denominator must not equal zero. Therefore, the denominator has a direct effect on the domain of the function. Restrictions on the domain result in vertical asymptotes or points of discontinuity on the graph of the function.

The parameters of the transformations in a rational function are similar to those of most other functions.

The equation of a transformed rational function is of the form y=ax−h+k.

a represents the vertical stretch factor and reflections across the horizontal axis.

h represents the horizontal translation.

k represents the vertical translation.

In the rational function y=2x−3+7, the parameters of transformation are as follows.

a=2
All the y-values of the base function y=1x will be multiplied by 2.
h=3
All the x-values will be translated 3 units to the right.
k=7
All the y-values will be translated 7 units up.

a.
Sketch the graph of the function y=4x+1+5 using transformations.
b.
What are the domain and range of the transformed function?

a.
Start with the graph of y=1x.

Note key points on the graph of the function.

There are points at 1,1,0.5,2,2,0.5, âˆ’1,−1,−0.5,−2 and −2,−0.5.

Parameters for y=4x+1+5 Resulting mapped point
a=4
There is a vertical stretch about the x-axis by a factor of 4. All y-values will be multiplied by 4.
1,1→1,40.5,2→0.5,82,0.5→2,2
h=−1
There is a translation to the left of 1 unit. All x-values will decrease by 1.
1,4→0,40.5,8→−0.5,82,2→1,2
k=5
There is a translation up of 5 units. All the y-values will increase by 5.
0,4→0,9−0.5,8→−0.5,131,2→1,7

The points mapped from quadrant III are as follows.

−1,−1→−1,−4→−2,−4→−2,1−0.5,−2→−0.5,−8→−1.5,−8→−1.5,−3−2,−0.5→−2,−2→−3,−2→−3,3

Now, plot the mapped points, and sketch the graph of the transformed function.

Note the horizontal asymptote has shifted up to y=5 from y=0.

Note also that there is a restriction on the domain of the transformed function.

x+1≠0x≠-1

There is a vertical asymptote at x=−1. The vertical asymptote for the transformed function shifted to the left 1 unit from that of y=1x.

b.
The domain of the transformed function y=4x+1+5 is x∣x≠−1,x∈R.

The range of the transformed function is y∣y≠5,y∈R.