1. 3.2 Graphing Quadratic Functions in Vertex or Intercept Form Definitions Definitions 3 Forms 3 Forms Steps for graphing each form Steps for graphing each form Examples Examples Changing between eqn. forms Changing between eqn. forms

2. Quadratic Function A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:

3. Vertex- The lowest or highest point of a parabola. Vertex Axis of symmetry- The vertical line through the vertex of the parabola. Axis of Symmetry

4. Vertex Form Equation y=a(x-h)2+k If a is positive, parabola opens up If a is negative, parabola opens down. The vertex is the point (h,k). The axis of symmetry is the vertical line x=h. Don’t forget about 2 points on either side of the vertex! (5 points total!)

5. Vertex Form  Each function we just looked at can be written in the form (x – h) 2 + k, where (h, k) is the vertex of the parabola, and x = h is its axis of symmetry.  (x – h) 2 + k – vertex form EquationVertex Axis of Symmetry y = x 2 or y = (x – 0) 2 + 0 (0, 0) x = 0 y = x 2 + 2 or y = (x – 0) 2 + 2 (0, 2) x = 0 y = (x – 3) 2 or y = (x – 3) 2 + 0 (3, 0) x = 3

6. Example 1: Graph y = (x + 2) 2 + 1 Analyze y = (x + 2) 2 + 1. Analyze y = (x + 2) 2 + 1. Step 1 Plot the vertex (-2, 1) Step 1 Plot the vertex (-2, 1) Step 2 Draw the axis of symmetry, x = -2. Step 2 Draw the axis of symmetry, x = -2. Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 3 Find and plot two points on one side, such as (-1, 2) and (0, 5). Step 4 Use symmetry to complete the graph, or find two points on the Step 4 Use symmetry to complete the graph, or find two points on the left side of the vertex. left side of the vertex.

7. Your Turn! Analyze and Graph: Analyze and Graph: y = (x + 4) 2 - 3. y = (x + 4) 2 - 3. (-4,-3)

8. Example 2: Graph y= -.5(x+3) 2 +4 a is negative (a = -.5), so parabola opens down. a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Axis of symmetry is the vertical line x = -3 Table of values x y Table of values x y -1 2 -1 2 -2 3.5 -2 3.5 -3 4 -3 4 -4 3.5 -4 3.5 -5 2 -5 2 Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

9. Now you try one! y=2(x-1) 2 +3 Open up or down? Open up or down? Vertex? Vertex? Axis of symmetry? Axis of symmetry? Table of values with 4 points (other than the vertex? Table of values with 4 points (other than the vertex?

10. (-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

11. Intercept Form Equation y=a(x-p)(x-q) The x-intercepts are the points (p,0) and (q,0). The x-intercepts are the points (p,0) and (q,0). The axis of symmetry is the vertical line x= The axis of symmetry is the vertical line x= The x-coordinate of the vertex is The x-coordinate of the vertex is To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. If a is positive, parabola opens up If a is positive, parabola opens up If a is negative, parabola opens down.

12. Example 3: Graph y=-(x+2)(x-4) Since a is negative, parabola opens down. The x-intercepts are (-2,0) and (4,0) To find the x-coord. of the vertex, use To find the y-coord., plug 1 in for x. Vertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) x=1 (-2,0)(4,0) (1,9)

13. Now you try one! y=2(x-3)(x+1) Open up or down? Open up or down? X-intercepts? X-intercepts? Vertex? Vertex? Axis of symmetry? Axis of symmetry?

14. (-1,0)(3,0) (1,-8) x=1

15. Changing from vertex or intercepts form to standard form The key is to FOIL! (first, outside, inside, last) The key is to FOIL! (first, outside, inside, last) Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 y=-x 2 +5x+36 =3(x 2 -2x+1)+8 =3x 2 -6x+3+8 =3x 2 -6x+3+8 y=3x 2 -6x+11 y=3x 2 -6x+11

16. Challenge Problem Write the equation of the graph in vertex form. Write the equation of the graph in vertex form.