Activity: TEKS: Overview: Materials: Grouping: Time: Square Roots & Quadratics: What s the Connection? (2A.9) Quadratic and square root functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: (G) connect inverses of square root functions with quadratic functions. In this activity, students will use their graphing calculators to investigate the connections between square root functions and quadratic functions. Students should be able to write the equations of functions by using transformation of functions. Students should also be familiar with the graph of quadratic and square root functions. Square Roots & Quadratics: What s the Connection Handout -1 per student Transparencies of the activity Graphing calculators Map colors 2 or 3 students 1 class period Lesson: Procedures 1. Divide students into groups of 2 or 3 and distribute the activity worksheets, calculators, and map colors. 2. Have students complete the activity worksheets in their groups. Allow 20 to 30 minutes for students to work in their groups. Notes Each student should receive his or her own set of worksheets. Each student needs to choose two different map color pencils for the activity. Circulate among the groups as they work to ensure that students remain on task and to answer questions as they arise. You may need to go over how to find the DrawInv function on their calculators and give guidance on finding an appropriate window to view both the function and its inverse. Refer to the Calculator Notes for Teachers pages at the end of this lesson if you need help with the DrawInv function on the calculator. Square Roots & Quadratics: What s the Connection? Page 1
Procedures Notes Detailed notes are written for examining problem i. You may want to have groups compare and discuss the first graph and their results to see that they are on the right track, or you may just want to walk the entire class through the analysis of the first graph. Guide students to make their sketches with enough detail to be able to identify ordered pairs especially the x- and y- intercepts of both graphs. Many students will try to copy the graph without marking either axis s units or labeling any ordered pairs. They will need these details for their work. 3. Have the groups share their results, and hold a class discussion of the observations that they have made through this investigation. Be sure to carefully guide the discussions, especially the results from (4) and (5). Discuss how to restrict the domain of a quadratic function in order to have an inverse that is a function. This is a good opportunity to introduce the term one-to-one function, and illustrate how, without a restriction on the domain, a quadratic function does not have an inverse that is a function. Emphasize that a square root function is only the inverse of a quadratic function if the domain is restricted appropriately (to one side of the vertex); likewise, a quadratic function is not the inverse of a square root function unless the domain is appropriately restricted. At the end of the activity, you can assist students in deducing a procedure on how you can find the equation of an inverse of a function (by switching the domain and range values of a function). Square Roots & Quadratics: What s the Connection? Page 2
Homework: Assign appropriate homework from the text, or provide some square root and quadratic functions and ask students to find their inverses. 1 Extensions: As an extension you can introduce the notation f ( x) You could also have them draw in the line y = x and explore the relationship between it and the graphs of a function and its inverse. Students could derive the geometric definition of inverse.. Resources: This activity was adapted from some investigations in Discovering Advanced Algebra, by Jerald Murdock, Ellen Kamischke, & Eric Kamischke, published by Key Curriculum Press. A good website for graph paper is http://www.mathematicshelpcentral.com/graph_paper.htm. Teachers may want to use a different template for the graph paper they use on this activity, and this is a good place to find other options. Square Roots & Quadratics: What s the Connection? Page 3
Square Roots & Quadratics: What s the Connection? Several square root and quadratic functions are given below: i. f ( x) = ( x + 6) 2 + 2 ii. f ( x) = 2 + x + 5 iii. f ( x) = 2 x f x =. 25x iv. ( ) 2 v. f ( x) = 6 + x 2 vi. f ( x) = ( x 2) 2 5 1. For each of these functions, do the following: a. Graph y 1 = f(x) on your calculator; then use the DrawInv function (under the DRAW menu) to draw the inverse of f on your calculator. Sketch both of these graphs on the graph grids provided (see following page) using one color to sketch the function and another color for its inverse. Make sure you label at least three points on both graphs. b. Determine whether the inverse is or is not a function. (Remember that a function passes the vertical line test.) Find the domain of the inverse, and write the domain on the lines provided to the right of each grid. Find an equation or equations for the inverse graphed and write them on the lines to the right of the grid. Verify your response by graphing the function in y 1 and graphing your equation of the inverse in y 2 on your calculator. Square Roots & Quadratics: What s the Connection? Page 4
i. ii. iii iv Square Roots & Quadratics: What s the Connection? Page 5
v. vi. Square Roots & Quadratics: What s the Connection? Page 6
2. Study your sketches and equations. What observations can you make? 3. Find the coordinates of the x-intercepts of each function above; then find the y-intercepts of the inverse. What to you notice? 4. Pair the functions given in (i) (vi) then explain your rationale for pairing them in this way. 5. Are any of these functions inverses of each other? Justify your answer. Square Roots & Quadratics: What s the Connection? Page 7
Teacher Solutions: Square Roots & Quadratics: What s the Connection? Several square root and quadratic functions are given below: f x =. 25x i. ( ) ( + 6) 2 2 f x = x + 2 iv. ( ) ii. f ( x) = 2 + x + 5 v. f ( x) = 6 + x 2 iii f ( x) = 2 x vi. f ( x) = ( x 2) 2 5 1. Blue graphs are functions. Red graphs are inverses. i. ii. Inverse is not a function Domain : x 2 Inverse y = 6± x 2 iii. iv. Inverse is a function Domain : x 0 Inverse : Inverse is not a function Domain : x 0 2 ( ) =.25 f x x y =± 2 x Square Roots & Quadratics: What s the Connection? Page 8
v. vi. Inverse is a function Domain : x 6 ( ) 2 f x = ( x+ 6) + 2 b. Determine whether the inverse is or is not a function. (Remember that a function passes the vertical line test.) Find the domain of the inverse, and write the domain on the lines provided on the right of each grid. Find an equation or equations for the inverse graphed and write them on the lines to the right of the grid. Verify your response by graphing the function in y 1 and graphing your equation of the inverse in y 2 on your calculator. When students write the equations for the inverse, if the inverse is a function students can use function notation to write its equation. If the inverse is not a function, then the equation needs to be written using y =. 2. Study your sketches and equations. What observations can you make? Answers will vary. They may notice that a parabola s inverse is not a function, unless you look at only a portion of its original graph. The inverse of a square root function is always a function and is only one half of a parabola. Students may also notice that the pattern in the coordinates of the inverse. Some of its points are related to the original function. In the original function a point has coordinates (x, y) and its inverse contains a point with coordinates (y, x) 3. Find the coordinates of the x-intercepts of each function above; then find the y- intercepts of the inverse. What to you notice? Answers will vary. Some possible answers. If the graph contains an x-intercept (a,0), then its inverse contains a y- intercept (0,a). If the original function does not have an x-intercept, its inverse does not have a y-intercept. Square Roots & Quadratics: What s the Connection? Page 9
4. Pair the functions given in (i) (vi) then explain your rationale for pairing them in this way. Answers may vary. i & v, ii & vi, and iii and iv. 5. Are any of these functions inverses of each other? Justify your answer. Answers may vary. If you look at the equations of the inverses for each square root function, they are half of a parabola function. If you look at the equations of the inverses for the parabolas, they are a pair of square root functions that each represents a different half of the original parabola graph. Square Roots & Quadratics: What s the Connection? Page 10
Calculator Notes to Get Started ( Problem i) Key strokes to draw the inverse. You have to input Y 1 first. The calculator just plots the points for the inverse of the function in Y 1. It does not find the equation of the inverse. Students can input equations in Y 2 that they think is the inverse to see if it matches the inverse drawn by the calculator. Students can change the way the calculator graphs their inverse function so that can compare their equation s graph with the drawing of the inverse. Square Roots & Quadratics: What s the Connection? Page 11
Students can see that this equation only matches half of the inverse. This inverse cannot be described by just one function. You may want to ask students what to try to find the equation for the bottom half of the function. Ask students why they think this happens. (From the drawing of the inverse of the original function students should have answered that it is not a function because it fails the vertical line test. This is the reason why the inverse cannot be expressed by a single function.) The inverse can be expressed as y = ± x 2 6. Before students graph the next function, students need to clear the drawing of the inverse that is currently on the screen by doing the following Square Roots & Quadratics: What s the Connection? Page 12