1. Graphing Parabolas Using the Vertex Axis of Symmetry & y-Intercept By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 15, 2007

2. Graphing Parabolas y = x 2 + 4x - 7 With your graphing calculator, graph each of the following quadratic equations and identify the vertex and axis of symmetry. y = 2x 2 + 10x + 4 y = -3x 2 + 5x + 9 Jeff Bivin -- LZHS

3. Graph the following parabola y = x 2 + 4x - 7 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

4. Graph the following parabola y = 2x 2 + 10x + 4 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

5. Graph the following parabola y = -3x 2 + 5x + 9 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

6. Graphing Parabolas y = x 2 + 4x - 7 Now look at the coefficients of the equation and the value of the axis of symmetry – especially a and b y = ax 2 + bx + c y = 2x 2 + 10x + 4 y = -3x 2 + 5x + 9 Jeff Bivin -- LZHS

7. Graphing Parabolas y = ax 2 + bx + c Vertex: Axis of symmetry: Jeff Bivin -- LZHS

8. Graph the following parabola y = x 2 + 4x - 7 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

9. Graph the following parabola y = 2x 2 + 10x + 4 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

10. Graph the following parabola y = -3x 2 + 5x + 9 axis of symmetry: vertex: y-intercept: Why did this parabola open downward instead of upward as did the previous two? Jeff Bivin -- LZHS

11. Graph the following parabola y = x 2 + 6x - 8 Axis of symmetry: Vertex: y-intercept: Jeff Bivin -- LZHS

12. Graph the following parabola y = -2x 2 + 7x + 12 Axis of symmetry: Vertex: y-intercept: Jeff Bivin -- LZHS

13. Graphing Parabolas In Vertex Form Jeff Bivin -- LZHS

14. Graphing Parabolas y = x 2 With your graphing calculator, graph each of the following quadratic equations and identify the vertex and axis of symmetry. y = (x - 5) 2 - 4 y = -3(x + 2) 2 + 5 y = ⅜(x - 3) 2 + 1 Jeff Bivin -- LZHS vertexaxis of sym.

15. Graph the following parabola y = (x - 5) 2 - 4 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

16. Graph the following parabola y = -3(x + 2) 2 + 5 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

17. Graph the following parabola y = ⅜(x - 3) 2 - 1 axis of symmetry: vertex: y-intercept: Jeff Bivin -- LZHS

18. Graphing Parabolas In Intercept Form Jeff Bivin -- LZHS

19. Graph the following parabola y = (x – 4)(x + 2) x-intercepts: vertex: y-intercept: Jeff Bivin -- LZHS axis of symmetry:

20. Graph the following parabola y = (x - 1)(x - 9) x-intercepts: vertex: y-intercept: Jeff Bivin -- LZHS axis of symmetry:

21. Graph the following parabola y = -2(x + 1)(x - 5) x-intercepts: vertex: y-intercept: Jeff Bivin -- LZHS axis of symmetry:

22. Convert to standard form y = -2(x + 1)(x - 5) Jeff Bivin -- LZHS y = -2(x 2 – 5x + 1x – 5) y = -2(x 2 – 4x – 5) y = -2x 2 + 8x + 10

23. Now graph from standard form. y = -2x 2 + 8x + 10 Axis of symmetry: Vertex: y-intercept: Jeff Bivin -- LZHS

24. A taxi service operates between two airports transporting 200 passengers a day. The charge is $15.00. The owner estimates that 10 passengers will be lost for each $2 increase in the fare. What charge would be most profitable for the service? What is the maximum income? Jeff Bivin -- LZHS Income = Price ● Quantity f(x) = ( 15 + 2x ) ( 200 – 10x ) Define the variable x = number of $2 price increases 15 + 2x = 0200 – 10x = 0 2x = -15 Vertex is: So, price = (15 + 2x) = (15 + 2(6.25)) = 15 + 12.5 = $27.50 f(x) = income 200 = 10x Maximum income: VERTEX 27.50 137.50

25. A taxi service operates between two airports transporting 200 passengers a day. The charge is $15.00. The owner estimates that 10 passengers will be lost for each $2 increase in the fare. What charge would be most profitable for the service? What is the maximum income? Jeff Bivin -- LZHS Income = Price ● Quantity f(x) = ( 15 + 2x ) ( 200 – 10x ) Define the variable x = number of $2 price increases f(x) = 3000 – 150x + 400x – 20x 2 f(x) = – 20x 2 + 250x + 3000 VERTEX f(6.25) = – 20(6.25) 2 + 250(6.25) + 3000 f(6.25) = 3781.25 Vertex is: So, price = (15 + 2x) = (15 + 2(6.25)) = 15 + 12.5 = $27.50 f(x) = income Maximum income = f(x) = $3781.25