5.1 Graphing Quadratic Functions (p. 249)

1. 5.1 Graphing Quadratic Functions(p. 249) • Definitions • 3 forms for a quad. function • Steps for graphing each form • Examples • Changing between eqn. forms

2. Quadratic Function • A function of the form y=ax2+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. Standard Form Equation y=ax2 + bx + c • If a is positive, u opens up If a is negative, u opens down • The x-coordinate of the vertex is at • To find the y-coordinate of the vertex, plug the x-coordinate into the given eqn. • The axis of symmetry is the vertical line x= • Choose 2 x-values on either side of the vertex x-coordinate. Use the eqn to find the corresponding y-values. • Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

5. 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!)

6. Intercept Form Equation y=a(x-p)(x-q) • The x-intercepts are the points (p,0) and (q,0). • The axis of symmetry is the vertical line x= • 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. • If a is positive, parabola opens up If a is negative, parabola opens down.

7. Example 1: Graph y=2x2-8x+6 • a=2 Since a is positive the parabola will open up. • Vertex: use b=-8 and a=2 Vertex is: (2,-2) • Axis of symmetry is the vertical line x=2 • Table of values for other points: x y • 0 6 • 1 0 • 2 -2 • 3 0 • 4 6 • * Graph! x=2

8. Now you try one!y=-x2+x+12* Open up or down?* Vertex?* Axis of symmetry?* Table of values with 5 points?

9. (.5,12) (-1,10) (2,10) (-2,6) (3,6) X = .5

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

11. Now you try one! y=2(x-1)2+3 • Open up or down? • Vertex? • Axis of symmetry? • Table of values with 5 points?

12. (-1, 11) (3,11) X = 1 (0,5) (2,5) (1,3)

13. Example 3: Graph y=-(x+2)(x-4) • The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) • 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) (1,9) (-2,0) (4,0) x=1

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

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

16. Changing from vertex or intercepts form to standard form • The key is to FOIL! (first, outside, inside, last) • Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8 =-(x2-9x+4x-36) =3(x-1)(x-1)+8 =-(x2-5x-36) =3(x2-x-x+1)+8 y=-x2+5x+36 =3(x2-2x+1)+8 =3x2-6x+3+8 y=3x2-6x+11

17. Assignment