Part I: Review of the Vertex Form of the Quadratic Function y = a(x h) 2 + k. 2 the graph becomes wider

Transcription

1 INVESTIGATING THE QUADRATIC FUNCTION IN STANDARD FORM y = ax 2 + bx + c Adapted from: Exploring Mathematics with the Transformation Graphing Application from Texas Instruments A quadratic function can be expressed in vertex form. You have already studied: y = a(x h) 2 + k where the point (h,k) is the vertex of the parabola. In this investigation you will study the standard form : y = ax 2 + bx + c. You will perform two investigations: 1. the changes on the graph of a quadratic function that will result from changes in a, b, and c 2. locate the vertex of a parabola given the equation in standard form Part I: Review of the Vertex Form of the Quadratic Function y = a(x h) 2 + k 1. What is the vertex of the parabola y = (x 3) 2 + 5? (3, 5) Does it open up or down? Up Now enter the equation in your graphing calculator and check your answer. 2. What happens to the graph if the equation is changed to y = 2(x 3) 2 + 5? the graph becomes skinnier Now enter the equation in your graphing calculator and check your answer 1 3. What happens to the graph if the equation is changed to y = (x 3) 2 + 5? 2 the graph becomes wider Now enter the equation in your graphing calculator and check your answer 4. What happens to the graph if the equation is changed to y = -5(x 3) 2 + 5? the graph becomes skinnier & opens down Now enter the equation in your graphing calculator and check your answer 5. What happens to the graph if the equation is changed to y = (x 3) 2 + 5? the graph becomes wider & opens down Now enter the equation in your graphing calculator and check your answer

2 6. What effect does changing the value of a in the vertex form of the function have on the vertex of a parabola? the width of the opening (skinny vs. wide) & the direction of the opening (up vs. down) 7. What are the coordinates of the vertex of the parabola y = 2(x + 3) 2 + 5? (-3, 5) 8. Use algebraic properties and rewrite the vertex form of the quadratic function to the standard form. 2(x+3)(x+3) + 5 2(x 2 + 6x + 9) + 5 2x x x x We already know that the x-coordinate of the function y = 2(x + 3) is -3 from the vertex form of the quadratic function. Now that the function has been changed to the standard form, what relationship exists between the x-coordinate of the vertex and values of a, b, and c in y = ax 2 + bx + c? (Note: If you do not see the relationship yet, the remainder of the activity will help to develop it.) Part II: Setting Up the Calculator to Investigate the Standard Form of the Quadratic Function y = ax 2 + bx + c In the calculator program we will be using, a is represented by A, b is represented by B, and c is represented by C so the standard form will look like: y = Ax 2 + Bx + C 1. Press APPS and select Transfrm and press ENTER.

3 2. Press any key to start the Transformation Graphing Application program. If you do not see the screen shown, select Continue. 3. Press Y= to display the Y = Editor. Clear any functions that are listed and turn off any plots. 4. In Y1 enter Ax 2 + Bx + C, the standard form of the quadratic equation. 5. Press WINDOW and the up arrow to display the SETTINGS screen for the Transformation Graphing Application. Enter the values shown. These settings define the starting values for the variables and the increment by which you want the variable values to change. Part III: Investigating the Effect of c y = ax 2 + bx + c In the calculator program we will be using, a is represented by A, b is represented by B, and c is represented by C so the standard form will look like: y = Ax 2 + Bx + C 1. Press ZOOM 6 to select 6:Standard and display the graph. The graph will show the pre-selected values of A, B, and C. Both the x and y axes will be set to display the graph between -10 and 10 with a scale of 1.

4 2. Press the down arrows to highlight C=. Press the right arrow to increase the value of C. What happens to the graph as the value of C increases? the vertex moves up 3. Press the left arrow to decrease the value of C. What happens to the graph as the value of C decreases? the vertex moves down 4. Use the up arrow key to highlight B= and change the value of B to any value. Highlight A= and then change the value of A to any positive value. Highlight C=. Increase and decrease the value of C. Is the effect on the graph different from what it was when A = 1 and B = 0? Explain your answer. no 5. When A equals one and B equals zero, what is the relationship between the vertex and the y- intercept? the vertex is the y intercept 6. Does the same relationship hold if A does not equal one or zero, but B equals zero? What if B does not equal zero? as long as B = 0, the vertex will be the y-intercept. however, if the value of B does not equal 0, then the vertex is not the y intercept 7. By knowing the value of C, what point on the graph do you know? the y-intercept 8. What is the y-intercept of a parabola if: C = 3? (0, 3) C = -1? (0, -1) C = -0.5? (0, -0.5) 9. Do the values of A and B affect the y-intercept of a parabola? no

5 Part IV: Investigating the Effects of a and b Together y = ax 2 + bx + c In the calculator program we will be using, a is represented by A, b is represented by B, and c is represented by C so the standard form will look like: y = Ax 2 + Bx + C 1. Reset the values of A, B, and C as shown. 2. Press 2 nd ZOOM. Using the cursor keys, highlight GridOn and press ENTER. With the GridOn, you can see the next segment? 3. Go to the WINDOW and change the settings as follows: 4. Press GRAPH to return to the graph screen.

6 5. Keeping A = 1, B = 0 and C = 0, increase the value of B. As B increases what happens to the location of the vertex of the parabola? As B increases, the vertex moves left and down.. 6. Keeping A = 1 and C = 0, set B = -6 and increase the value of B. As B increases what happens to the location of the vertex of the parabola? As B increases, the vertex moves left and up. 7. Keeping A = 1 and C = 0, set B = 0 and decrease the value of B. As B decreases what happens to the location of the vertex of the parabola? As B decreases, the vertex moves right and down 8. Keeping A = 1 and C = 0, set B = 6 and decrease the value of B. As B decreases what happens to the location of the vertex of the parabola? As B decreases, the vertex moves right and up.

7 9. Did the y-intercept change as B changed? no 10. Use the Transformation Graphing Application program and change the value of B to fill in the table below. Value of B Value of A x-coordinate of the vertex What relationship do you notice between the value of the x-coordinate of the vertex and the value of B when A remains 1? The x-coordinate is half of B, when A = On your graphing calculator, set A = 2, B = 0 and C = 0. When the value of A is changed from 1 to 2, what happens to the graph of the parabola? Does the vertex change? the graph becomes narrower 13. Use the Transformation Graphing Application program to change the value of B and complete the table below. Value of B Value of A x-coordinate of the vertex

8 14. Now investigate the effect of doubling the value of A on the x-coordinate of the vertex of a parabola. Use the Transformation Graphing Application program to change the values of A and B and fill in the table below. Value of B x-coordinate when A = 1 x-coordinate of when A = What is the effect on the x-coordinate of the vertex when the value of A is doubled from 1 to 2? the x-coordinate is cut in half 16. With A equal to a number other than one, what relationship do you notice between the value of the x-coordinate and the value of B? This will be the answer to question number 9 in part II of this lesson. Use the expression you wrote to check each x-coordinate value from number 14. -(B/A)/2 = x coordinate CHECK: 17. Use your expression to complete the table below. Value of B Value of A x-coordinate of the vertex