Chapter 10 Quadratic Equations & Functions

1. Chapter 10Quadratic Equations & Functions Examine Graphs & Their Equations Solve Quadratic Equations by Graphing, Factoring, & Using the Quadratic Formula

2. Section 10 – 1 Exploring Quadratic Equations Objectives: To graph quadratic functions of the form

3. Standard Form of a Quadratic Function Examples:

4. Parabola: The U-Shaped Curve the graph of a quadratic function makes.

5. Axis of Symmetry: The line (sometimes invisible) that divides the parabola into two matching halves.

6. Vertex: The highest or lowest point of the parabola, found on the axis of symmetry. (The bottom or top of the U).

7. Minimum Vs. Maximum When the Parabola Opens UP (like a U), the vertex is at the bottom. We call this a minimum. When the Parabola Opens DOWN (like an A), the vertex is at the top. We call this a maximum.

8. Minimum Vs. Maximum (U) Minimum: When a > 0 (Positive) (A) Maximum: When a < 0 (Negative)

9. Example 1 Identifying a Vertex Identify the vertex of each graph. Tell whether it is a minimum or maximum. A) B)

10. C) D)

11. Example 2 Graphing Make a table of values and graph each quadratic function. A) Functions in the form have a vertex at (0, 0)! Plot a few points on one side of the vertex, and then reflect each point across the axis of symmetry!

12. B) Functions in the form have a vertex at (0, 0)! Plot a few points on one side of the vertex, and then reflect each point across the axis of symmetry!

13. C) Functions in the form have a vertex at (0, 0)! Plot a few points on one side of the vertex, and then reflect each point across the axis of symmetry!

14. Example 3 Comparing Widths of Parabolas A) Use the graphs below. Order the quadratic functions from widest to narrowest graph.

15. B) Use the graphs below. Order the quadratic functions from widest to narrowest graph.

16. C)Order the quadratic functions from widest to narrowest graph.

17. D)Order the quadratic functions from widest to narrowest graph.

18. Homework Textbook Page 513; #1 – 13 All (Use Graph Paper for #4 – 9)

20. 1) Name the vertex of each parabola, then determine whether it is a minimum or maximum. Warm Up 2) Order the quadratic functions from widest to narrowest graph.

21. Section 10 – 1 Continued… Objectives: To graph quadratic functions of the form

22. The value of C, the constant term in a quadratic function, translates the graph up or down!

23. Example 4Graphing Graph each quadratic function. Compare the Graphs. A) and

24. Graph each quadratic function. Compare the Graphs. B) and

25. Real- World Connection You can model the height of an object moving under the influence of gravity using a quadratic function. As an object falls, its speed continues to increase. Ignoring air resistance, you can approximate height of a falling object using a function . The height (h) is in feet, the time (t) is in seconds, and the initial height of the object (c) is in feet.

26. Example 5Graphing A) Suppose you see an eagle flying over a canyon. The eagle is 30 feet above the level of the canyon’s edge when it drops a stick from its claws. The force of gravity causes the stick to fall toward the Earth. The function gives the height of the stick (h) in feet after (t) seconds. Graph this quadratic function.

27. B)Suppose a squirrel is in a tree 24 feet above the ground. She drops an acorn. The function give the height of the acorn in feet after t seconds. Graph this function.

28. Homework • 10 – 1 Ditto; 2 – 30 Even