Daily Check Factor: 3x2 + 10x + 8 Factor and Solve: 2x2 - 7x + 3 = 0

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:

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

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!)

Vertex Form (x – h)2 + k – 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 Equation Vertex Axis of Symmetry y = x2 or y = (x – 0)2 + 0 (0 , 0) x = 0 y = x2 + 2 or y = (x – 0)2 + 2 (0 , 2) y = (x – 3)2 or y = (x – 3)2 + 0 (3 , 0) x = 3

Example 1: Graph y = (x + 2)2 + 1 Analyze y = (x + 2)2 + 1. 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.

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

Example 2: Graph y= -.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

Table of values with 4 points (other than the vertex? Now you try one! 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) (3,11) X = 1 (0,5) (2,5) (1,3)

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.

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) (1,9) (-2,0) (4,0) x=1

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

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

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

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