Module 1
Part 7: Functions TRANSFORM!!!!
OBJ: Students will be able to transform graphs of functions
EVI: Students will be able to solve problems that transform graphs
Vertical and Horizontal Shifts
Let c be a positive real number. Vertical and Horizontal shifts in the graph of 𝑦 = 𝑓(𝑥) are represented
as follows.
Vertical shift c units upward
ℎ(𝑥) = 𝑓(𝑥) + 𝑐
Vertical shift c units downward
ℎ(𝑥) = 𝑓(𝑥) − 𝑐
Horizontal shift c units to the right
ℎ(𝑥) = 𝑓(𝑥 − 𝑐)
Horizontal shift c units to the left
ℎ(𝑥) = 𝑓(𝑥 + 𝑐)
Reflections in the Coordinate Axes
Reflections in the coordinate axes of the graph of 𝑦 = 𝑓(𝑥) are represented as follows
Reflection in the x-axis
ℎ(𝑥) = −𝑓(𝑥)
Reflection in the y-axis
ℎ(𝑥) = 𝑓(−𝑥)
Stretching or Shrinking
Non-rigid transformations are those that cause a distortion in the original shape.
Vertical stretch or shrink
𝑔(𝑥) = 𝑐𝑓(𝑥) stretch if 𝑐 > 1 and shrink if 0 < 𝑐 < 1
Horizontal stretch or shrink
𝑔(𝑥) = 𝑓(𝑐𝑥) horizontal shrink if 𝑐 > 1 and a horizontal stretch if 0 < 𝑐 < 1
Describe the following transformations
A. 𝑔(𝑥) = 𝑥 3 − 1
B. ℎ(𝑥) = (𝑥 + 2)3 + 1
Using the graph 𝒇(𝒙) = 𝒙𝟒 do the following transformations
1. Reflected across the x-axis followed by an upward shift of two units
2. Horizontal shift of three units to the right followed by a reflection in the y-axis
Compare the graph of each function with the graph of 𝒇(𝒙) = |𝒙|
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A. 𝑔(𝑥) = 3|𝑥|
B. 𝑔(𝑥) = 3 |𝑥|
Compare the graph of each function with the graph of 𝒇(𝒙) = 𝟐 − 𝒙𝟑
A. 𝑔(𝑥) = 𝑓(2𝑥)
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B. 𝑔(𝑥) = 𝑓 (2 𝑥)
Module 1
Part 7: Functions TRANSFORM!!!!
OBJ: Students will be able to transform graphs of functions
EVI: Students will be able to solve problems that transform graphs
HOMEWORK!!!!
In Exercises 1-8 g is related to one of the parent functions described previously in part 6. (a) Identify the
parent function f. (b) Describe the sequence of transformations from f to g.
1. 𝑔(𝑥) = (𝑥 − 8)2
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2. 𝑔(𝑥) = 3 𝑥 2 + 4
3. 𝑔(𝑥) = 2(𝑥 − 7)2
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4. 𝑔(𝑥) = √4 𝑥
5. 𝑔(𝑥) = −|𝑥| − 2
6. 𝑔(𝑥) = 2⟦𝑥 + 5⟧
7. 𝑔(𝑥) = √𝑥 + 4 + 8
8. 𝑔(𝑥) = √7 − 𝑥 − 2
In Exercises 9-13, write an equation for the function described by the given characteristics
9. The shape of 𝑓(𝑥) = 𝑥 2 , but moved two units to the right and eight units downward
10. The shape of 𝑓(𝑥) = 𝑥 2 , but moved three units left, seven units upward, and reflected in the xaxis
11. The shape of 𝑓(𝑥) = 𝑥 3 , but moved six units to the left, six units downward, and reflected
across the y-axis
12. The shape of 𝑓(𝑥) = |𝑥|, but moved 10 units upward and reflected in the x-axis
13. The shape of 𝑓(𝑥) = √𝑥, but moved six units to the left and reflected in both the x-axis and the
y-axis