Mastering Quadratic Functions: Graphing, Modeling & Symmetry

1. Ch 2.2 Characteristics of Quadratic Functions

2. Standard Form of a parabola f(x) = ax²+bx + c

4. EX 1: Graph f(x) = 3x² -6x + 1. Label the vertex and axis of symmetry Step 1: Identify the coefficients. Because a > 1, the parabola opens ______ Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the x-coordinate of the vertex Another good point is the Identify the y-intercept c up a=3 b = -6 c = 1

5. Step 1: Identify the coefficients. Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the x-coordinate of the vertex Another good point is the Identify the y-intercept c You Try!

6. EX 2: Modeling with Quadratics A softball player hits a ball whose path is modeled by where x is the distance from home plate (in feet) and y is the height of the ball above the ground (in feet). What is the highest point this ball will reach? If the ball was hit to center field which has an 8 foot fence located 410 feet from home plate, was this hit a home run? Explain.

7. You Try! The height h (in feet) of water spraying from a fire hose can be modeled by h(x) = −0.03x² + x + 25, where x is the horizontal distance (in feet) from the fire truck. What is the highest point the water reaches?

8. What is “Intercept Form” of a quadratic function?

9. EX 3: How can we graph a quadratic function in Intercept Form? Graph f(x) = −2(x + 3)(x − 1). Label the x-intercepts, vertex, and axis of symmetry down Step 1: Identify and plot the x-intercepts. Because a < 1, the parabola opens ______ Step 2: Find the vertex Step 3: Draw the axis of symmetry Step 4: Make a table centered around the vertex, Which includes the x-intercepts p = -3 and q = 1

10. You Try! Graph f(x) = (x + 1)(x − 3) Label the x-intercepts, vertex, and axis of symmetry Graph f(x) = − (x -4)(x + 2). Label the x-intercepts, vertex, and axis of symmetry

11. EX 3: Modeling with Quadratics The parabola shows the path of your first golf shot, where x is the horizontal distance (in yards) and y is the corresponding height (in yards). The path of your second shot can be modeled by the function f(x) = −0.02x(x − 80). Which shot travels farther before hitting the ground? Which travels higher?

12. You Try! The flight of a particular gold shot can be modeled by the function y = -.001x(x-260) where x is the horizontal distance (in yards) and y is the corresponding height (in yards). Which shot travels farther before hitting the ground? Which travels higher?

13. SUMmary! Essential Question: What types of symmetry does the graph of a quadratic function have? Describe this symmetry. Other Questions to consider: If you know the vertex of a parabola, can you graph the parabola? If you knew the vertex and one additional point on the graph, would that be enough to graph the parabola?

14. Classwork: This question, pg 61 # 15-19all, 13, 31,53, 49 (extra practice pg 66 #11-14 all)

15. Pg 66

16. Sorting Activity Instructions You are given 16 graphs and 32 equations. Each graph needs two equations pasted under it: vertex form and standard form The easiest way to do this is to check the vertex and y-intercepts! Standard Form:

19. HINT: Look for the “y” intercept” which is the value c in y= ax² + bx + cOR foil vertex form and simplify Standard Form: Standard Form:

20. Standard Form Standard Form