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

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:

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

Standard Form Equation 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. 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 2 x-values on either side of the vertex x- coordinate. Use the eqn to find the corresponding y- values. 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. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

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 yTable of values for other points: x y * Graph! x=2

Now you try one! y=-x 2 +x+12 * Open up or down? * Vertex? * Axis of symmetry? * Table of values with 5 points?

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

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!)

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

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 5 points? Table of values with 5 points?

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

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.

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)

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?

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

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