4.3 Graph Quadratic Functions in Vertex or Intercept Forms

1. 4.3 Graph Quadratic Functions in Vertex or Intercept Forms

2. Standard Form of a Quadratic ax2 + bx + c = 0 Find the axis of symmetry, state the vertex, and how it is concaved. -3x – x2 + 4 = 8 AOS = x = -1.5 Vertex = (-1.5, -1.75) Concaved Down

3. Review: Graph of Vertex Form

4. Example 1: Graph a Quadratic Function in Vertex Form Graph Step 1: Identify the constants (a, h, and k) a = -1/4 h = -2 k = 5 Remember that (h, k) is your vertex!! Step 2: Plot the vertex and draw the axis of symmetry

5. 2 y = -1/4(x + 2) + 5 2 y = -1/4(0 + 2) + 5 2 y = -1/4(2) + 5 y = -1/4(4) + 5 y = -1+ 5 Example 1: Graph a Quadratic Function in Vertex Form Graph Step 3: Evaluate the function for one value of x. Choose an easy one!! For example, use x = 0! Plot (0, 4) y = 4

6. Example 1: Graph a Quadratic Function in Vertex Form Graph Step 4: Reflect the point over the axis of symmetry. What do you get? (-4, 4) Step 5: Draw a parabola through your points!

7. Your Turn! Graph the function. Label the vertex and axis of symmetry. Vertex: (-2, -3) AOS: x = -2

8. Use a Quadratic Model in Vertex Form The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function where x and y are measured in feet. What is the distance d between the two towers?

9. Use a Quadratic Model in Vertex Form SOLUTION What is the vertex? (1400, 27) What does that mean? Remember that the vertex is in the middle of the parabola That means that each tower is 1400 feet from the midpoint. So…? The distance between the two towers is 2(1400) or 2800ft.

10. Graph of Factored Form y = a(x – p)(x – q)

11. Step 1: Identify the x-intercepts. Remember it’s the opposite of what you think!! If (x + 3), then p = ______ If (x – 1), then q = ______ Example 3: Graph a Quadratic Function in factored Form Graph -3 1 Plot the intercepts!!!

12. Example 3: Graph a Quadratic Function in factored Form Graph Step 2:Find the x-coordinate of the vertex. = -1 Step 3:Find the y-coordinate of the vertex. y = 2(x + 3)(x – 1) y = 2(-1 + 3)(-1 – 1) y = 2(2)(-2) y = -8 So…plot the vertex (-1, -8) and connect the points!

13. 2 y = -2(x - 8x + 5x – 40) 2 y = -2(x - 3x – 40) 2 y = -2x + 6x + 80 Example 4: Change from Factored to Standard Form Write y = -2(x + 5)(x – 8) in standard form. 1. Write original function y = -2(x + 5)(x – 8) 2. Multiply using FOIL orDouble Distribution 3. Combine like terms 4. Distribute the -2 to get standard form

14. 2 Write f(x) = 4(x - 1) +9 in standard form. 2 f(x) = 4(x – 1) + 9 f(x) = 4(x – 1)(x – 1) + 9 2 2. Rewrite (x – 1) 2 f(x) = 4(x - 1x – 1x + 1) + 9 Example 5: Change from Vertex to Standard Form 1. Write original function 3. Multiply using FOIL orDouble Distribution

15. 2 Write f(x) = 4(x - 1) +9 in standard form. 2 2 f(x) = 4(x - 2x + 1) + 9 f(x) = (4x - 8x + 4) + 9 f(x) = 4x - 8x + 13 2 Example 5: Change from Vertex to Standard Form 4. Combine like terms 5. Distribute the 4 6. Combine like terms to get standard form

16. Re-write the equation in Standard Form: y = 3(x – 1)(x + 2)

17. Re-write the equation in Standard Form: y = -1(x + 3)2 – 4