9.1.2 Parabola Investigation Do Now 1. Vertical means and horizontal is. 2. Another word for compress is. 3. Given the statement 0 < a < 1, a represents numbers like 4. Given the statement a > 1, a represents numbers like 5. 0 2 =, 1 2 =, 4 2 =, 9 2 =, 10 2 = 6. The important points of a parabola are. 7. The vertex is the point on the graph of a parabola. Notes Rules for Transformations of Functions If f x is the original functions, a > 0 and c > 0. Transformation Name Function Transformations of the graph of f x Open downward f(x) Reflect about the Vertical Shift f x + c Shifts c units f x c Shifts c units Horizontal Shift f(x + c) Shifts to the c units f(x c) Shifts to the c units Stretch Vertically a f x, a > 1 vertically by a factor of a Stretch Horizontally f ax, a > 1 horizontally by a factor of!! Compress Vertically a f x, 0 < a < 1 vertically by a factor of a Compress Horizontally f ax, 0 < a < 1 horizontally by a factor of!! Problems Highlight important information and circle the prompt(s)/ question(s). In Chapter 5 you learned how to graph parabolas and solve quadratic equations. In this lesson you will develop more tools for graphing parabolas with particular characteristics. 9-12. PARABOLA LAB: Polly Parabola has been the manager of the Parabola Department of Functions of America, but she has decided to start her own company called Professional Parabola Productions. She needs your help. See her memo below. To: Your Study Team From: Ms. Polly Parabola Re: New Parabola Possibilities I am starting a new company specializing in parabolas. To win over new customers, I need to be able to show them that we know more about parabolas than any of the other function factories around, especially since every company already sells f(x) = x 2. My customers will need all sorts of parabolas, and we need the knowledge to make the customers happy. I 1
would love to offer parabolas that are completely new to them. Please investigate all different kinds of parabolas. Determine all the ways that you can change the equation f(x) = x 2 to change the shape, direction, and location of its graph. Remember that I m counting on you! I need you to uncover parabola secrets that our competitors do not know. Sincerely, Ms. Polly Parabola 9-13. Make a complete graph of the parabola f(x) = x 2. Be sure to label any important points. When you are sure that your graph is complete and accurate, trace over it in colored pencil. (Hint: make a table then graph it!) a. How can you change the equation to make the f(x) = x 2 parabola stretch vertically? (That is, to make the graph look narrower, so the points in the parabola seem to rise away from the vertex more quickly. The new parabola should have the same vertex and orientation (i.e., opens upward) as f(x) = x 2.) Record the equations you try along with their graphs. 2
b. How can you change the equation to make the f(x) = x 2 parabola compress vertically? (That is, to make the graph look wider so that the points seem to rise away from the vertex less quickly.) Record the equations you try, along with their graphs and your observations. c. How can you change the equation to make the same parabola open downward? (The new parabola should be congruent to f(x) = x 2, with the same vertex, but it should open downward so its vertex will be its highest point.) Record the equations you try, their graphs, and your observations. 3
d. How can you change the equation to shift the f(x) = x 2 parabola 5 units down? (Your new parabola should look exactly like f(x) = x 2, but the vertex should be at (0, 5).) Record the equations you try, along with their graphs. Describe how to shift the graph up as well as down. e. How can you change the equation to shift the f(x) = x 2 parabola 3 units to the right? (Your new parabola should look exactly like f(x) = x 2, but the vertex should be at the point (3, 0).) Record the equations you try, along with their graphs. Describe how to shift the parabola to the left as well as how to shift it to the right. 4
f. How can you change the equation to shift the f(x) = x 2 parabola 3 units to the left, as in part (e), AND stretch vertically, as in part (a)? Record the equations you try, along with their graphs. 9-14. How can you change the equation f(x) = x 2 to make the parabola vertically compressed, open downward, shifted six units up, and shifted two units to the left? Where is the vertex of the new parabola? (Hint: transform f(x) = x 2 notice where the vertex is located after all transformations are performed.) 5
9-15. Now that you are a parabola expert, you can impress Ms. Polly Parabola! Make up your own fancy transformation and show her how you can change the equation f(x) = x 2 to create it. Answers should indicate how each parameter in the equation relates to the transformation. 9-16. Use what you learned in the Parabola Lab to write an equation for each of the parabolas described below. (Hint: transform f(x) = x 2 ) a. A parabola opening upward, shifted 7 units right, and 4 units down. f(x) = (x 7) 2 4 b. A parabola that is vertically stretched by a factor of 2, sitting with its vertex on the x-axis at x = 3. f(x) = 2(x + 3) 2 c. A downward-opening parabola with vertex ( 5, 2) and a vertical compression of 0.5. f(x) = 0.5(x + 5) 2 + 2 Homework Sec 9.1.2 #9-17, 9-18, 9-19, 9-20, 9-22 6