1. 5.1 Graphing Quadratic Functions Do now: Make up three examples of linear functions. How do you know they are linear? OBJ: to graph quadratic functions & use quadratic functions to solve real-life problems

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

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- y=ax 2 + bx + c y=ax 2 + bx + c If a is positive, u opens up If a is positive, u opens up If a is negative, u opens down The x-coordinate of the vertex is at 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= The axis of symmetry is the vertical line x= Choose an x-value & use the eqn to find the corresponding y-value. Then use symmetry to find another point Choose an x-value & use the eqn to find the corresponding y-value. Then use symmetry to find another point Graph and label the points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve. Graph and label the points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

5. Ex 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 yTable of values for other points: x y 06 06 2-2 2-2 * Graph! x=2

6. Graphing calculator 2 nd 2 nd CALC CALC Option 3: Min or Option 4: Max Option 3: Min or Option 4: Max Left bound? use your arrows until you are to the left of the vertex Left bound? use your arrows until you are to the left of the vertex Enter Enter Right bound? use your arrows until you are to the right of it Right bound? use your arrows until you are to the right of it Enter Enter Use 2 nd TABLE to find two other points Use 2 nd TABLE to find two other points

7. Practice: Graph y = -x 2 + x + 12 * Opens up or down? * What is the Vertex? * Where is the Axis of symmetry? * Table of values with points & use symmetry

8. (-1,10) (-2,6) (2,10) (3,6) X =.5 (.5,12)

9. Vertex Form- 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!

10. Ex 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 -3 4 -3 4 -5 2 -5 2 Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

11. Graphing calculator 2 nd 2 nd TABLE TABLE Find the “mirror” - this is the vertex Find the “mirror” - this is the vertex Then choose 2 points - one on either side Then choose 2 points - one on either side

12. Practice: Graph y = 2(x - 1) 2 + 3 *Opens up or down? *Vertex? *Axis of symmetry? *Table of values with points

13. (-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

14. Intercept Form- y = a(x-p)(x-q) If a is positive, parabola opens up If a is positive, parabola opens up If a is negative, parabola opens down. 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.

15. Ex 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=1The axis of symmetry is the vertical line x=1 x=1 (-2,0)(4,0) (1,9)

16. Graphing calculator 2 nd 2 nd CALC CALC Option 2: “ zeros ” Option 2: “ zeros ” Left bound? use your arrows until you are to the left of the x-int Left bound? use your arrows until you are to the left of the x-int Enter Enter Right bound? use your arrows until you are to the right of it Right bound? use your arrows until you are to the right of it Enter Enter REPEAT for the other x-int (zero) REPEAT for the other x-int (zero)

17. Practice: Graph y = 2(x - 3)(x + 1) *Open up or down? *x-intercepts?*Vertex? *Axis of symmetry?

18. (-1,0)(3,0) (1,-8) x=1

19. Changing from vertex or intercepts form to standard form The key is to use the distributive property The key is to use the distributive property Ex 4: y=-(x+4)(x-9)Ex 5: y=3(x-1) 2 +8 =-(x 2 -9x+4x-36) =3(x-1)(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 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