1. 5.1/ 5.2 Graphs of Quadratic Functions Components of the graph and Vertex form

2. A little parabola exploration We’ll start looking at quadratics by looking at a parabola. What’s the parent function for parabolas? Use the CAS document to play and find patterns.

3. The “Bits” General form: y = ax 2 + bx +c If a > 0, then the graph “smiles.” If a < 0, then it “frowns.” c is the y-intercept. Vertex: the high or low point-- also known as the maximum or minimum point. Axis of symmetry: the line the parabola is centered on. For these “smiling” and “frowning” parabolas, the axis will be what type of line? Symmetric point: the point of the y-intercept reflected over the axis of symmetry. Find the vertex, axis of symmetry, and symmetric point for y = x 2 + 2x + 2.

4. General/ standard form Find the vertex, axis of symmetry, and symmetric point for y = x 2 + 2x + 2. What utility on your calculator could you use to determine the vertex/ minimum?

5. Vertex form You can use the vertex form to find the vertex algebraically. Here is the form: y - k = a(x - h) 2 The vertex is the point (h, k). Find the vertex by rewriting a standard equation into vertex form. Do this by completing the square.

6. Completing the Square Let’s do this to demonstrate the method. y = x 2 - 6x + 2 Subtract 2 from each side: y - 2 = x 2 - 6x Square the half of -6 and add to each side: y - 2 + 9 = x 2 - 6x +9 Combine terms and rewrite as a square: y + 7 = (x - 3) 2 What is the vertex? What are the y-intercept and symmetric point?

7. Completing the Square How should these graphs compare? y = x 2 - 6x + 2 y + 7 = (x - 3) 2 What is the vertex? What are the y-intercept and symmetric point?

8. Completing the Square Let’s do this to demonstrate the method when a ≠ 1. y = 3x 2 - 24x + 17 Subtract 17 from each side: y - 17 = 3x 2 - 24x Factor out the 3. y - 17 = 3(x 2 - 8x) Square the half of -8 and add to each side. Don’t forget to account for that 3:y - 17 + 48 = 3(x 2 -8x +16) Combine terms and rewrite as a square: y + 31 = 3(x - 4) 2 What is the vertex? What are the y-intercept and symmetric point?

9. Completing the Square Again, the graphs should be exactly the same. y = 3x 2 - 24x + 17 y + 31 = 3(x - 4) 2 What is the vertex? What are the y-intercept and symmetric point?