Do Now: 1.Find the axis of symmetry: 2. See page 176 and do #19 Student will be able to transform a quadratic equation in standard form to vertex form.

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

 The vertex from the calculator can help put it in vertex form! Example quadratic equation: (Vertex)

 The vertex from the calculator can help put it in vertex form!

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

y=a(x-h) 2 +k IIf a is positive, parabola opens up If a is negative, parabola opens down. TThe vertex is the point (h,k). TThe axis of symmetry is the vertical line x=h.

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

 Analyze y = (x + 2)  Step 1 Plot the vertex (-2, 1)  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 4 Use symmetry to complete the graph, or find two points on the  left side of the vertex.

 Complete the square: y = x 2 +8x + 13 Subtract the 13 Add (½ b) 2 to both sides finish complete the square

 Y – 13 = x 2 +8x  Y = x 2 +8x + 16 y +3 = (x + 4) 2 Y = (x + 4) (-4,-3)

 Analyze and Graph: y = (x + 4) (-4,-3)

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

Y=2x 2 -4x +5  Subtract the 5  Factor out the 2  Use complete the square

 Y=2x 2 -4x +5  Y-5 =2x 2 -4x  Y-5+___ =2(x 2 -2x+___)  Y = 2(x 2 – 2x +1) (why add 2 on the left?)  Y-3 = 2(x – 1) 2 (now add 3 to both sides….

y=2(x-1) 2 +3  Open up or down?  Vertex?  Axis of symmetry?  Table of values with 4 points (other than the vertex?

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

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.

SSince a is negative, parabola opens down. TThe x-intercepts are (-2,0) and (4,0) TTo find the x-coord. of the vertex, use TTo find the y-coord., plug 1 in for x. V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)

y=2(x-3)(x+1)  Open up or down?  X-intercepts?  Vertex?  Axis of symmetry?

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

 The key is to FOIL! (first, outside, inside, last)  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 -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 y=3x 2 -6x+11

 Write the equation of the graph in vertex form.

page , 34-37