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TEKS (4)(C) Determine the effect on the graph of f ( x ) = 1 x when f ( x ) is replaced by af ( x ), f ( x ) + d , f ( bx ), and f ( x - c ) for specific positive and negative values of a , b , c , and d. TEKS (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations , including symbols, diagrams, graphs, and language as appropriate.
TEKS FOCUS
ฤ Square root parent function^ โ The square root parent function is the simplest form of the square root function, or f ( x ) = 1 x.
ฤ Implication โ a conclusion that follows from previously stated ideas or reasoning without being explicitly stated ฤ Representation^ โ a way to display or describe information. You can use a representation to present mathematical ideas and data.
VOCABULARY
The graph of any square root function is a transformation of the graph of the square root parent function, f ( x ) = 1 x.
ESSENTIAL UNDERSTANDING
Parent Function f ( x ) = 1 x , x ร 0
Vertical Translation Horizontal Translation y = 1 x + d y = 1 x - c
d 7 0: shifts up 0 d 0 units c 7 0: shifts to the right 0 c 0 units
d 6 0: shifts down 0 d 0 units c 6 0: shifts to the left 0 c 0 units
Vertical Stretch and Compression Horizontal Stretch and Compression y = a 1 x y = 1 bx
0 a 0 7 1: vertical stretch 0 b 0 7 1: horizontal compression (shrink)
0 a 0 6 1: vertical compression (shrink) 0 b 0 6 1: horizontal stretch
a 6 0: reflection in x -axis b 6 0: reflection in y -axis
Key Concept Square Root Function Family
Transformations of Square Root Functions
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Problem 3P bl 3
Problem 2P bl 2
Problem 1P
Vertical Stretch and Compression
What are the graphs of y = 3 1 x, y =^12 1 x, and y = โ 2 1 x? The graph of y = 31 x is the graph of y = 1 x stretched vertically by a factor of 3. The graph of y = 12 1 x is the graph of y = 1 x compressed vertically by a factor of 12. The graph of y = - 21 x is the graph of y = 1 x stretched vertically by a factor of 2 and reflected in the x -axis. The domains of all three functions are the set of nonnegative numbers, but their ranges vary.
TEKS Process Standard (1)(D)
x O (^) 5 10
y
y = 3 x
y = โ 2 x
y = x
y =^12 x
Translating a Square Root Function Vertically
What are the graphs of y = 1 x โ 2 and y = 1 x + 1? The graph of y = 1 x - 2 is the graph of y = 1 x shifted down 2 units. The graph of y = 1 x + 1 is the graph of y = 1 x shifted up 1 unit. The domains of both functions are the set of nonnegative numbers, but their ranges differ.
TEKS Process Standard (1)(D)
y 2
O x
y x
y x 1
y x 2
Translating a Square Root Function Horizontally
What are the graphs of y = 1 x + 4 and y = 1 x โ 1? The graph of y = 1 x + 4 is the graph of y = 1 x shifted left 4 units. The graph of y = 1 x - 1 is the graph of y = 1 x shifted right 1 unit. The ranges of both functions are the set of nonnegative numbers, but their domains differ.
TEKS Process Standard (1)(D)
y
4 2 O 2 4 x
y x y^ ^ x^ ^^1
y x 4
W
TT
v T v T v T
How is y = a 1 x related to the parent function f(x) = 1 x? If 0 a 0 7 1, it is a vertical stretch by a factor of 0 a 0. If 0 a 0 6 1, it is a vertical compression by a factor of 0 a 0. If a 6 0, it is also a reflection in the x-axis.
W
TT
d T 1 T n
How is y = 1 x + d related to the parent function y = 1 x? It is related to the parent function in the same way that y = f (x) + d is related to y = f (x). It is a vertical translation of d units.
TT
T
n
How is y = 1 x โ c related to the parent function y = 1 x? It is a horizontal translation of c units.
244 Lesson 6-3 Transformations of Square Root Functions
PRACTICE and APPLICATION EXERCISES
ONLINE HO MEWORK
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Graph each transformation of the parent function f (x) = 1 x. Analyze the effect of the transformation on the graph of the parent function.
1. y = 1 x + 1 2. y = 1 x - 2 3. y = 1 x - 4 4. y = 1 x + 5 5. y = 1 x - 3 6. y = 1 x + 1 7. y = 1 x + 6 8. y = 3 1 x 9. Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Suppose that a function pairs elements from set A with elements from set B. Recall that a function is called onto if every element in B is paired with at least one element in A.
a. The graph shows a transformation of y = 1 x. Write the function. b. What are the domain and range of the function? c. For the domain, is the function onto the set of nonnegative real numbers? Explain.
10. Write a transformation of the parent square root function such that for its domain, the function is onto the set of real numbers such that y โฆ 3. 11. a. Graph y = 1 - x , y = 11 - x , and y = 12 - x. b. Analyze Mathematical Relationships (1)(F) How does the graph of y = 1 c - x differ from the graph of y = 1 x - c? 12. How is the graph of y = 1 x - 5 translated from the graph of y = 1 x? A. shifted 5 units left C. shifted 5 units up B. shifted 5 units right D. shifted 5 units down Graph each transformation of the parent function f (x) = 1 x. Analyze the effect of the transformation on the graph of the parent function. 13. y = 14 1 x 14. y = - 21 x 15. y = 16 x 16. y = 5 13 x 17. y = 1 - 5 x 18. y = 5 - 23 x 19. y = 12 x + 1 20. y = 31 x + 2
y
x O (^) 2 2
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246 Lesson 6-3 Transformations of Square Root Functions
Write a square root function matching each description.
21. The parent function f ( x ) = 1 x is compressed vertically by a factor of 101 , translated 4 units down, and reflected in the x -axis. 22. The parent function f ( x ) = 1 x is compressed horizontally by a factor of 7.5 and translated 2 units up. 23. The parent function f ( x ) = 1 x is translated 12 unit left and stretched vertically by a factor of 3. 24. The parent function f ( x ) = 1 x is stretched vertically by a factor of 10, translated 5 units down, and reflected in the y -axis. 25. Evaluate Reasonableness (1)(B) A company makes steel food cans of different sizes. All of the cans are 10 cm tall, but their radii vary. The equation r = 0.18 1 V gives the radius of a can based on the canโs volume. a. Describe this equation as a transformation of y = 1 x. b. The volume of one size of can is 300 cubic centimeters. What is the radius of this can? Round to the nearest hundredth. c. Explain how you can check to see if your answer is reasonable.
Write the function shown in each graph.
26. 27.
y
x O 5 10
y
x O 5 10
y
x O
y
x O 5
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