COMMON CORE 4 9 7 5 Locker LESSON Graphing Square Root Functions Common Core Math Standards The student is epected to: COMMON CORE F-IF.C.7 Graph square root... functions. Also F-IF.B.4, F-IF.B.6, F-BF.B. Mathematical Practices COMMON CORE. MP.4 Modeling Language Ojective Discuss with a partner how the graphs of square root functions compare with quadratic functions. ENG AGE Essential Question: How can ou use transformations of a parent square root function to graph functions of the form f () = a - h + k or g () = ( - h) + k? Possile answer: You can use the parameters a,, h, and k to transform points on the parent function and use those transformed points to draw the graph of f () or g (). Houghton Mifflin Harcourt Pulishing Compan Name Class Date. Graphing Square Root Functions Essential Question: How can ou use transformations of a parent square root function to graph Eplore Graphing and Analzing the Parent Square Root Function Although ou have seen how to use imaginar numers to evaluate square roots of negative numers, graphing comple numers and comple valued functions is eond the scope of this course. For purposes of graphing functions ased on the square roots (and in most cases where a square root function is used in a real-world eample), the domain and range should oth e limited to real numers. The square root function is the inverse of a quadratic function with a domain limited to positive real numers. The quadratic function must e a one-to-one function in order to have an inverse, so the domain is limited to one side of the verte. The square root function is also a one-to-one function as all inverse functions are. The domain of the square root function (limited to real numers) is given ǀ Fill in the tale. 4 9 f () = Plot the points on the graph, and connect them with a smooth curve. Recall that this function is the inverse of the parent quadratic (ƒ () = ) with a domain limited to the nonnegative real numers. Write the range of this square root function: ǀ functions of the form g () = a (-h) + k or g () = The graph appears to e getting flatter as increases, indicating that the rate of change as increases. 7 5 (-h) + k? - 5 7 decreases Resource Locker PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how a representative sample can give information aout the entire atch. Then preview the Lesson Performance Task. Descrie the end ehavior of the square root function, ƒ () = _. ƒ () as Module 495 Lesson Name Class Date. Graphing Square Root Functions Essential Question: How can ou use transformations of a parent square root function to graph functions of the form g () = a (-h) + k or g () = F-IF.C.7 Graph square root... functions. Also F-IF.B.4, F-IF.B.6, F-BF.B. Eplore Graphing and Analzing the Parent Houghton Mifflin Harcourt Pulishing Compan Square Root Function Although ou have seen how to use imaginar numers to evaluate square roots of negative numers, graphing comple numers and comple valued functions is eond the scope of this course. For purposes of graphing functions ased on the square roots (and in most cases where a square root function is used in a real-world eample), the domain and range should oth e limited to real numers. The square root function is the inverse of a quadratic function with a domain limited to positive real numers. The quadratic function must e a one-to-one function in order to have an inverse, so the domain is limited to one side of the verte. The square root function is also a one-to-one function as all inverse functions are. ǀ Plot the points on the graph, and connect them with a smooth curve. The domain of the square root function (limited to real numers) is given Fill in the tale. f () = Recall that this function is the inverse of the parent quadratic (ƒ () = ) with a domain limited to the nonnegative real numers. Write the range of this square root function: ǀ (-h) + k? Resource - 5 7 The graph appears to e getting flatter as increases, indicating that the rate of change as increases. decreases Descrie the end ehavior of the square root function, ƒ () = _. ƒ () as Module 495 Lesson HARDCOVER PAGES 55 64 Turn to these pages to find this lesson in the hardcover student edition. 495 Lesson.
Reflect. Discussion Wh does the end ehavior of the square root function onl need to e descried at one end? The function is not defined for negative values of, and so its ehavior as approaches negative infinit is also not defined. The discrete point (, ) does not descrie end ehavior even though the function ends there. EXPLORE Graphing and Analzing the Parent Square Root Function. The solution to the equation = 4 is sometimes written as = ±. Eplain wh the inverse of ƒ () = cannot similarl e written as g () = ± _ in order to use all reals as the domain of ƒ (). Functions, including functions that are inverses of other functions, must have a single range value for each domain value. The ± smol is shorth and for positive or negative and indicates two different range values for a single domain value. This is counter to the definition of function. Eplore Predicting the Effects of Parameters on the Graphs of Square Root Functions You have learned how to transform the graph of a function using reflections across the - and -aes, vertical and horizontal stretches and compressions, and translations. Here, ou will appl those transformations to the graph of the square root function ƒ () = _. When transforming the parent function ƒ () = _, ou can get functions of the form g () = a ( - h) + k or g () = ( - h) + k. For each parameter, predict the effect on the graph of the parent function, and then confirm our prediction with a graphing calculator. A Predict the effect of the parameter, h, on the graph of g () = _ - h for each function. a. g () = _ - : The graph is a translation of the graph of ƒ () [right/left/up/down] units.. g () = _ + : The graph is a translation of the graph of ƒ () [right/left/up/down] units. Check our answers using a graphing calculator. B Predict the effect of the parameter k on the graph of g () = _ + k for each function. a. g () = _ translation + : The graph is a of the graph of ƒ () [right/up/left/down] units.. g () = _ translation - : The graph is a of the graph of ƒ () [right/up/left/down] units. Check our answers using a graphing calculator. Module 496 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions Students have learned how parameters affect the graphs of quadratic functions, asolute value functions, and rational functions. In this lesson, these concepts are etended to the graphs of square root functions. Students analze how the parameters a,, h, and k, in functions of the form f () = a - h + k and f () = ( - h) + k affect the graph of the parent square root function, f () =. The then use this knowledge to graph square root functions, and to analze graphs to determine the functions the represent. Houghton Mifflin Harcourt Pulishing Compan INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activit either in the ook or online. How do ou know that the graph of f () = does not have a horizontal asmptote? As the value of increases, the square root of increases without ound, so there is no horizontal asmptote. (As, f ().) The function is an increasing function, and the range is all non-negative real numers. EXPLORE Predicting the Effects of Parameters on the Graphs of Square Root Functions For functions of the form f () = a, wh does a value of a that is etween and create a vertical compression of the graph of the parent function? The function value for each -value is multiplied a numer less than, making it less than the corresponding value for the parent function; this pulls the graph closer to the -ais. Graphing Square Root Functions 496
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Discuss with students how the can use what the know aout how the various parameters affect the graphs of quadratic functions to make predictions aout how the parameters will affect the graphs of square root functions. C Predict the effect of the parameter a on the graph of g () = a _ for each function. a. g () = _ : The graph is a vertical stretch of the graph of ƒ () a factor of. _. g () = _ : The graph is a vertical compression of the graph of ƒ () a factor of. c. g () = - _ : The graph is a vertical _ compression of the graph of ƒ () a factor of as well as a reflection across the -ais. d. g () = - _ : The graph is a vertical stretch of the graph of ƒ () a factor of as well as a reflection -ais across the. Check our answers using a graphing calculator. D Predict the effect of the parameter,, on the graph of g () = for each function. a. g () = horizontal : The graph is a stretch of the graph of ƒ () a factor of. _. g () = _ : The graph is a horizontal compression of the graph of ƒ () a factor of. c. g () = _ - _ : The graph is a horizontal stretch of the graph of ƒ () a factor of as well as a reflection across the -ais. _ d. g () = _ - : The graph is a horizontal compression of the graph of ƒ () a factor of as well as a reflection across the -ais. Check our answers using a graphing calculator. Reflect Houghton Mifflin Harcourt Pulishing Compan. Discussion Descrie what the effect of each of the transformation parameters is on the domain and range of the transformed function. Each value in the domain shifts left h if h is positive and right h if h is negative. h has no effect on the range. If a is negative, it changes the sign of each value in the range, ut a has no effect on the domain. Each value in the range shifts up k if k is positive and down k if k is negative. k has no effect on the domain. If is negative, it changes the sign of each value in the domain, ut has no effect on the range. Module 497 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activit Have students work in pairs. Have each student write a function of the form f () = a - h + k ut keep it hidden from the partner. Have students graph their functions on graph paper, echange them with partners, and tr to determine the function represented the partner s graph. Have partners check each other s work. Then have the students repeat the activit using functions of the form f () = _ ( - h ) + k. 497 Lesson.
Eplain Graphing Square Root Functions When graphing transformations of the square root function, it is useful to consider the effect of the transformation on two reference points, (, ) and (, ), that lie on the parent function, and where the map to on the transformed function, g (). f () = g () = a - h + k g () = ( - h) + k h k h k h + k + a h + k + The transformed reference points can e found recognizing that the initial point of the graph is translated from (, ) to (h, k). When g () involves the parameters a, h, and k, the second transformed reference point is unit to the right of (h, k) and a units up or down from (h, k), depending on the sign of a. When g () involves the parameters, h, and k, the second transformed reference point is units left or right from (h, k), depending on the sign of, and unit aove ( h, k ). Transformations of the square root function also affect the domain and range. In order to work with real valued inputs and outputs, the domain of the square root function cannot include values of that result in a negative-valued epression. Negative values of can e in the domain, as long as the result in nonnegative values of the epression that is inside the square root. Similarl, the value of the square root function is positive definition, ut multipling the square root function a negative numer, or adding a constant to it changes the range and can result in negative values of the transformed function. Eample g () = _ - - For each of the transformed square root functions, find the transformed reference points and use them to plot the transformed function on the same graph with the parent function. Descrie the domain and range using set notation. To find the domain: Square root input must e nonnegative. - Solve the inequalit for. The domain is ǀ To find the range: The square root function is nonnegative. _ - Multipl _ - Sutract. _ - - - The epression on the left is g (). g () - Since g () is greater than or equal to - for all in the domain, 6 4 Houghton Mifflin Harcourt Pulishing Compan EXPLAIN Graphing Square Root Functions What is an eample of a negative value of that is in the domain of a square root function? Wh is it part of the domain? In the function g () = +, can e -6 ecause it results in a nonnegative square root, g () = -6 + = 4. What effect do h and k have on the domain and range of f() = a - h + k? The domain is all real numers greater than or equal to h. The range is all real numers greater than or equal to k if a >, and less than or equal to k if a <. INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Students can use a graphing calculator to help them etter understand the effects of a stretch or a compression. Have them graph oth the parent function and the given function on the calculator so the can see how the rate of change of the function is reflected in the graph. the range is ǀ - (, ) (, -) (, ) (4, ) - 4 6 8 Module 498 Lesson DIFFERENTIATE INSTRUCTION Kinesthetic Eperience Have students work in pairs. Have each pair place a transparenc sheet over a sheet of graph paper with the aes laeled. Have them graph the radical function f () = a - h + k on the transparenc for h = and k =. Then have them move the transparenc to represent changes in h and k, and write the function that represents the transformation. Students can use a graphing calculator to check their answers. Graphing Square Root Functions 498
PEER-TO-PEER DISCUSSION Ask students to discuss with a partner the difference etween a vertical stretch and a horizontal stretch. Have them use graphs ehiiting each of these attriutes to descrie the comparison in their own words. Then ask them to do the same for vertical and horizontal compressions. Have them share their descriptions with the class. B g () = - _ ( - ) + To find the domain: Square root input must e nonnegative. - _ ( - ) Multipl oth sides -. - Add to oth sides. Epressed in set notation, the domain is To find the range: The square root function is nonnegative. - _ ( - ) Add oth sides - _ ( - ) + Sustistute in g (). g () Since g () is greater than for all in the domain, the range (in set notation) is ǀ (, ) (, ) (, ) (, ) - - - Your Turn For each of the transformed square root functions, find the transformed reference points and use them to plot the transformed function on the same graph with the parent function. Descrie the domain and range using set notation. 4. g () = - _ - + (, ) (, ) (, ) (, ) Houghton Mifflin Harcourt Pulishing Compan - - 5 7 To find the domain: - The domain is ǀ To find the range: - - - - - + g () The range is ǀ Module 499 Lesson LANGUAGE SUPPORT Communicate Math Have students work in pairs to complete a chart similar to the following, comparing and contrasting quadratic and square root functions. Tpe of Function Eample Description Similarities Differences 499 Lesson.
5. g () = - Eplain ( + ) + Writing Square Root Functions Given the graph of a square root function and the form of the transformed function, either g () = a _ - h + k or g () = ( - h) = k, the transformation parameters can e determined from the transformed reference points. In either case, the initial point will e at (h, k) and readil apparent. The parameter a can e determined how far up or down the second point (found at = h + ) is from the initial point, or the parameter can e determined how far to the left or right the second point (found at = k + ) is from the initial point. Eample - - Write the function that matches the graph using the indicated transformation format. g () = ( - h) + k Initial point: (h, k) = (,-) - - - - Second point: (, ) (-, ) (, ) (, ) To find the domain: _ ( + ) + (h +, k +) = (, -) + = = - The function is g () = - ( - ) -. - The domain is ǀ - To find the range: _ _ ( + ) ( + ) + g () The range is ǀ Houghton Mifflin Harcourt Pulishing Compan EXPLAIN Writing Square Root Functions How can ou tell the signs of h and k looking at the graph? You can see which wa the graph of the parent function has een translated. If it has een translated to the right, h is positive. If it has een translated to the left, h is negative. If it has een translated up, k is positive. If it has een translated down, k is negative. AVOID COMMON ERRORS Students ma forget that when indicating a horizontal stretch or compression of the parent graph, the reciprocal of the stretch or compression factor, and not the factor itself, is placed under the radical sign. Present a side--side comparison of a stretch and a compression to remind students of the proper procedure. Module 5 Lesson Graphing Square Root Functions 5
INTEGRATE TECHNOLOGY Students can use a graphing calculator to check their work. The can enter their functions and use the graphing feature to check that the graphs of their functions match the given graph. B g () = a _ - h + k 6 4 4 Initial point: (h, k) = ( -, - ) Second point: (h +, k + a ) = (-, ) - + a = a = The function is g () = + -. Your Turn Write the function that matches the graph using the indicated transformation format. 6. g () = _ ( - h) + k 7. g () = a ( - h) + k - (h, k) = (, -) (h +, k + ) = ( _, ) + = _ = _ g () = ( - ) - Houghton Mifflin Harcourt Pulishing Compan 7 5 - - (h, k) = (-, 5) (h +, k + a) = (-, ) 5 + a = a = - g () = - _ + + 5 Module 5 Lesson 5 Lesson.
Eplain Modeling with Square Root Functions Square root functions that model real-world situations can e used to investigate average rates of change. Recall that the average rate of change of the function ƒ () over an interval from to is given ƒ ( ) - ƒ ( ) -. EXPLAIN Modeling with Square Root Functions Eample Use a calculator to evaluate the model at the indicated points, and connect the points with a curve to complete the graph of the model. Calculate the average rates of change over the first and last intervals and eplain what the rate of change represents. The approimate period T of a pendulum (the time it takes a pendulum to complete one swing) is given in seconds the formula T =. _ l, where l is the length of the pendulum in inches. Use lengths of, 4, 6, 8, and inches. First find the points for the given -values. Length (inches) Period (seconds).45 4.64 6.78 8.9. Plot the points and draw a smooth curve through them. Find the average increase in period per inch increase in the pendulum length for the first interval and the last interval. First interval: rate of change = _.64 -.45 4 - =.95 Last Interval: rate of change = _. -.9-8 =.5 The average rate of change is less for the last interval. The average rate of change represents the increase in pendulum period with each additional inch of length. As the length of the pendulum increases, the increase in period time per inch of length ecomes less. Period (seconds)..8.6.4. T 4 6 8 Length (inches) l Houghton Mifflin Harcourt Pulishing Compan. Olga Mishna/ Shutterstock How can ou summarize the rate of change of a square root function of the form f () = a when a is a positive numer? How is this reflected in the graph of the function? As values of increase, the rate of change decreases. The graph shows a rising curve that gets less and less steep (i.e., flatter) as approaches infinit. Module 5 Lesson Graphing Square Root Functions 5
B A car with good tires is on a dr road. The speed, in miles per hour, from which the car can stop in a given distance d, in feet, is given s (d) = _ 96d. Use distances of, 4, 6, 8, and feet. First, find the points for the given -values. Distance 4 6 8 Speed 4.8 6. 75.9 87.6 98. Plot the points and draw a smooth curve through them. s First interval: 6. - 4.8 rate of change = 4 - =.9 Last Interval: 98. - 87.6 rate of change = - 8 =.5 The average rate of change is less for the last interval. The average rate of change represents the increase in speed with each additional foot of distance. As the availale stopping distance increases, the additional increase in speed per foot of stopping distance decreases. Speed (mi/h) 8 6 4 4 6 8 Distance (feet) d Houghton Mifflin Harcourt Pulishing Compan Your Turn Use a calculator to evaluate the model at the indicated points, and connect the points with a curve to complete the graph of the model. Calculate the average rates of change over the first and last intervals and eplain what the rate of change represents. 8. The speed in miles per hour of a tsunami can e modeled the function s (d) =.86 d, where d is the average depth in feet of the water over which the tsunami travels. Graph this function from depths of feet to 5 feet and compare the change in speed with depth from the shallowest interval to the deepest. Use depths of,,, 4, and 5 feet for the -values. Points: (,.), (, 7.6), (,.4), (4, 44.), (5, 7.9) s First interval: = 7.6 -. - 4 =.55 Last Interval: = 7.9-44. 5-4 8 =.88 The average rate of change is less for the last interval. The average rate of change represents the increase in 6 tsunami speed with each additional foot of depth. The speed increases less with each additional foot of depth. 4 Speed (mi/h) Depth (feet) d Module 5 Lesson 5 Lesson.
Elaorate 9. What is the difference etween the parameters inside the radical ( and h) and the parameters outside the radical (a and k)? The inside parameters are horizontal transformations and the outside parameters are vertical transformations.. Which transformations change the square root function s end ehavior? vertical reflections (a < ) and horizontal reflections ( < ). Which transformations change the square root function s initial point location? horizontal translations (h ) and vertical translations (k ). Which transformations change the square root function s domain? horizontal translations (h ) and horizontal reflections (a < ). Which transformations change the square root function s range? vertical translations (k ) and vertical reflections ( < ) ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP. Enhance students understanding of the different effects produced a and having them complete a tale of values for =,, 4, 6, and 64 and the functions f () =, g () = 4, h () = 4, and, and compare the values across the 4 functions. Students can then plot the points and graph the different functions to see how the effects of the parameters are reflected in the graphs. j () = 4. Essential Question Check-In Descrie in our own words the steps ou would take to graph a function of the form g () = a - h + k or g () = ( - h) + k if ou were given the values of h and k and using either a or. Start with the point (h, k), which is the endpoint of the function. If g () = a - h + k is used, a second reference point will e located over to the right and up a (or down a if a < ). If g () = ( - h) + k is used, the second reference point is found moving up and to the right (or left if < ). Draw a half paraola that goes through oth reference points and continues past the second point. Houghton Mifflin Harcourt Pulishing Compan How can ou use the rule for a square root function to determine the -intercept of its graph? Find the value of that makes the function equal to. This can e found inspection, or setting the rule equal to and solving for. SUMMARIZE THE LESSON - h + k related to the graph of f () =? Its starting point is at (h, k), instead of (, ). It is stretched verticall a factor of a if a >, or compressed verticall a factor of a if < a <. If a <, then the graph is decreasing instead of increasing. How is the graph of f () = a Module 54 Lesson Graphing Square Root Functions 54