Unit 3

Transformations

Combining Transformations on a Radical Function

In Lesson 3.1, you performed more than one transformation on a given function. The process you learned can be applied to specific functions.

The equation of a transformed radical function is of the form y=abx−h+k. In this equation, a is the vertical stretch factor, 1b is the horizontal stretch factor, h is the horizontal translation, and k is the vertical translation. If a is a negative, there will be a reflection in the x-axis. If b is negative, there will be a reflection in the y-axis.

Sketch the graph of y=−2x+3−4 by transforming the graph of y=x.

State the domain and range of the transformed function.

Start by sketching the graph of y=x, and pick some key points for mapping to the image function.
The points 0,0 and 4,2 can be used for mapping.

The function y=−2x+3−4 corresponds to y=−2fx+3−4.

The values of a,b,h, and k are as follows

a=−2b=1h=−3k=−4

The a-value is negative, so there is a reflection in the x-axis. All y-values will change to the opposite sign.

0,0→0,04,2→4,−2

Because a=2, there is a vertical stretch about the x-axis by a factor of 2. All y-values will be multiplied by 2.

0,0→0,04,−2→4,−4

Because b=1, there is no horizontal stretch or reflection.

Because h=−3, there is a horizontal shift 3 units to the left. All x-values will decrease by 3.

0,0→−3,04,−4→1,−4

Because k=−4, there is a vertical shift 4 units down. All y-values will decrease by 4.

−3,0→−3,−41,−4→1,−8
The domain of the transformed function is x∣x≥−3,x∈R.

The range is y∣y≤−4,y∈R.