5.1 Graphing Quadratic Functions

1. 5.1 Graphing Quadratic Functions

2. What is a Quadratic Function? A function with the form y = ax 2 + bx + c where a  0. The graph is U-shaped and is called a parabola .

3. Why Study Quadratics? Many things we see every day are modeled by quadratic functions. What are some examples? Water in a drinking fountain The McDonald’s Golden Arches The path of a basketball

4. Quadratic Vocab The lowest or highest point is called the vertex . The axis of symmetry is a vertical line through the vertex.

5. Effects of “a” Standard Form: y = ax 2 + bx + c Just like with absolute value functions: If a > 0 (+), the parabola opens up If a < 0 (-), the parabola opens down If |a| < 1, the parabola is wider than y = x 2 If |a| > 1, the parabola is narrower than y = x 2

6. Finding the Vertex The x-coordinate of the vertex is To find the y, plug in the x-coordinate. Example: Find the vertex. y = 2x 2 – 8x + 6 Axis of symmetry is the line x =

7. Graphing in Standard Form Find and plot the vertex. Draw the axis of symmetry. Choose two x values on one side and plot the points. Use symmetry to plot two points on the other side. Connect the points with a curve (parabola). Example: y = 2x 2 – 8x + 6

8. Example: Graph y = -x 2 + 4x - 2

9. Your Turn! Graph

10. Vertex Form y = a(x – h) 2 + k The vertex is (h, k) Axis of symmetry is x = h. Just like absolute value functions: h shifts right or left k shifts up or down a determines direction and width

11. Graphing in Vertex Form Same steps as for Standard Form. Example: Graph

12. Example: Graph

13. Your Turn! Graph

14. Intercept Form y = a(x – p)(x – q) X-intercepts are p and q Axis of symmetry is halfway between (p, 0) and (q, 0).

15. Graphing in Intercept Form Plot the intercepts: (p, 0) and (q, 0) The x-coordinate of the vertex is in the middle of the x-intercepts. Plug it in to find y, then plot the vertex. Connect the 3 points with a curve (parabola). Example: y = -(x + 2)(x – 4)

16. Example: Graph

17. Your Turn! Graph

18. Writing in Standard Form Multiply! (Use FOIL) Example: Write in standard form:

19. Example: Write in standard form:

20. Your Turn! Write in standard form: 1. 2.