6-3. Transformations of Square Root Functions. Key Concept Square Root Function Family VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING
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1 -3 Transformations of Square Root Functions TEKS FOCUS TEKS ()(C) Determine the effect on the graph of f() = when f() is replaced b af(), f() + d, f(b), and f( - c) for specific positive and negative values of a, b, c, and d. TEKS ()(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. VOCABULARY Square root parent function The square root parent function is the simplest form of the square root function, or f() =. Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING The graph of an square root function is a transformation of the graph of the square root parent function, f () =. Ke Concept Square Root Function Famil Parent Function f () =, Ú 0 Vertical Translation = + d d 7 0: shifts up 0 d 0 units d 0: shifts down 0 d 0 units Horizontal Translation = - c c 7 0: shifts to the right 0 c 0 units c 0: shifts to the left 0 c 0 units Vertical Stretch and Compression Horizontal Stretch and Compression = a = b a 7 : vertical stretch b 7 : horizontal compression (shrink) a : vertical compression (shrink) b : horizontal stretch a 0: reflection in -ais b 0: reflection in -ais PearsonTEXAS.com 3
2 Problem TEKS Process Standard ()(D) How is = a function f() =? If 0 a 0 7, it is a vertical stretch b a factor of 0 a 0. If 0 a 0, it is a vertical compression b a factor of 0 a 0. If a 0, it is also a reflection in the -ais. Vertical Stretch and Compression What are the graphs of = 3, =, and =? The graph of = 3 is the graph of = stretched verticall b a factor of 3. The graph of = is the graph of = compressed verticall b a factor of. The graph of =- is the graph of = stretched verticall b a factor of and reflected in the -ais. The domains of all three functions are the set of nonnegative numbers, but their ranges var. = 3 8 = = O = Problem TEKS Process Standard ()(D) Translating a Square Root Function Verticall How is = + d function =? It is function in the same wa that = f () + d is related to = f (). It is a vertical translation of d units. What are the graphs of = and = +? The graph of = - is the graph of = shifted down units. The graph of = + is the graph of = shifted up unit. The domains of both functions are the set of nonnegative numbers, but their ranges differ. O 8 Problem 3 TEKS Process Standard ()(D) Translating a Square Root Function Horizontall What are the graphs of = + and =? How is = c function =? It is a horizontal translation of c units. The graph of = + is the graph of = shifted left units. The graph of = - is the graph of = shifted right unit. The ranges of both functions are the set of nonnegative numbers, but their domains differ. O Lesson -3 Transformations of Square Root Functions
3 Problem Horizontal Stretch and Compression How is =!b function f() =!? If 0 b 0 7, it is a horizontal compression b a factor of 0 b 0. If 0 b 0, it is a horizontal stretch b a factor of 0 b 0. If b 0, it is also a reflection in the -ais. What are the graphs of =, = 53, and = 3? The graph of = is the graph of = compressed horizontall b a factor of. The graph of = 53 is the graph of = stretched horizontall b a factor of 3. The graph of = -3 is the graph of = compressed horizontall b a factor of 3 and reflected in the -ais. The ranges of all three functions are the set of nonnegative numbers, but their domains var. 8 = = 3 = O = Problem 5 Graphing a Square Root Function What would be good points to choose? Points that have integer - and -coordinates. What is the graph of = 3 +? Step Step Step 3 Choose several points from the parent function =. Multipl the -coordinates b a =-. This shrinks the parent graph verticall b the factor and reflects the result in the -ais. The values of c and d give the horizontal and vertical translations. Translate the graph from Step to the right 3 units and up unit. O!!! 3 Step Step 3 Step PearsonTEXAS.com 5
4 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Graph each transformation of the parent function f () =. Analze the effect of the transformation on the graph of the parent function. For additional support when completing our homework, go to PearsonTEXAS.com.. = +. = - 3. = -. = = - 3. = + 7. = + 8. = 3 9. Use Multiple Representations to Communicate Mathematical Ideas ()(D) Suppose that a function pairs elements from set A with elements from set B. Recall that a function is called onto if ever element in B is paired with at least one element in A. O a. The graph shows a transformation of =. 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? Eplain. 0. 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 3.. a. Graph = -, = -, and = -. b. Analze Mathematical Relationships ()(F) How does the graph of = c - differ from the graph of = - c?. How is the graph of = - 5 translated from the graph of =? 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 () =. Analze the effect of the transformation on the graph of the parent function. 3. =. =- 5. =. = = = = + 0. = 3 + Lesson -3 Transformations of Square Root Functions
5 Write a square root function matching each description.. The parent function f () = is compressed verticall b a factor of 0, translated units down, and reflected in the -ais.. The parent function f () = is compressed horizontall b a factor of 7.5 and translated units up. 3. The parent function f () = is translated unit left and stretched verticall b a factor of 3.. The parent function f () = is stretched verticall b a factor of 0, translated 5 units down, and reflected in the -ais. 5. Evaluate Reasonableness ()(B) A compan makes steel food cans of different sizes. All of the cans are 0 cm tall, but their radii var. The equation r = 0.8V gives the radius of a can based on the can s volume. a. Describe this equation as a transformation of =. 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. Eplain how ou can check to see if our answer is reasonable. Write the function shown in each graph.. 7. O 5 0 O O - -5 O PearsonTEXAS.com 7
6 30. Appl Mathematics ()(A) The qualit control supervisor at a car part factor uses the equation = to determine the number of parts,, to inspect based on the number manufactured,. a. Describe this equation as a transformation of =. b. The supervisor determined that 55 parts should be inspected. How man were manufactured? 3. Eplain Mathematical Ideas ()(G) Use equations to eplain wh a vertical stretch b a factor of 3 is the same as a horizontal compression b a factor of 9. TEXAS Test Practice 3. Which of the following functions translates the graph of f () = up 3 units and left 7 units? A. = C. = B. = D. = Which of the following best describes the transformation of =-5 from f () =? F. horizontal stretch b factor of 5 H. vertical stretch b factor of 5 and reflection in -ais and reflection in -ais G. horizontal compression b factor J. vertical compression b factor of 5 of 5 and reflection in -ais and reflection in -ais 3. Which function has a domain of Ú? A. = + C. = + B. = - D. = In which quadrant of the coordinate plane is the graph of =--? F. Quadrant I H. Quadrant III G. Quadrant II J. Quadrant IV 3. What is the -intercept of = + + 3? Eplain using transformations of this function from the parent function, f () =. 8 Lesson -3 Transformations of Square Root Functions
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