1. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Warm Up Give the coordinate of the vertex of each function. 2. f(x) = 2(x + 1) 2 – 4 1. f(x) = (x – 2) 2 + 3 3. Give the domain and range of the following function. (2, 3) (–1,–4) {(–2, 4), (0, 6), (2, 8), (4, 10)} D:{–2, 0, 2, 4}; R:{4, 6, 8, 10}

2. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems. Objectives

3. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form This shows that parabolas are symmetric curves. The axis of symmetry is the line through the vertex of a parabola that divides the parabola into two congruent halves.

4. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Example 1: Identifying the Axis of Symmetry Identify the axis of symmetry for the graph of.

5. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Another useful form of writing quadratic functions is the standard form. The standard form of a quadratic function is f(x)= ax 2 + bx + c, where a ≠ 0.

6. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form These properties can be generalized to help you graph quadratic functions.

7. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 – 4x + 5. Example 2A: Graphing Quadratic Functions in Standard Form a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept.

8. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 – 4x + 5. Example 2A: Graphing Quadratic Functions in Standard Form e. Graph the function.

9. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Consider the function f(x) = –x 2 – 2x + 3. Example 2B: Graphing Quadratic Functions in Standard Form a. Determine whether the graph opens upward or downward. b. Find the axis of symmetry. c. Find the vertex. d. Find the y-intercept.

10. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form e. Graph the function. Consider the function f(x) = –x 2 – 2x + 3.

11. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form

12. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Find the minimum or maximum value of f(x) = –3x 2 + 2x – 4. Then state the domain and range of the function. Example 3: Finding Minimum or Maximum Values

13. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form The highway mileage m in miles per gallon for a compact car is approximately by m(s) = –0.025s 2 + 2.45s – 30, where s is the speed in miles per hour. What is the maximum mileage for this compact car to the nearest tenth of a mile per gallon? What speed results in this mileage? Example 4

14. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Lesson Quiz: Part I 1. Determine whether the graph opens upward or downward. 2. Find the axis of symmetry. 3. Find the vertex. 4. Identify the maximum or minimum value of the function. 5. Find the y-intercept. Consider the function f(x)= 2x 2 + 6x – 7.

15. Holt McDougal Algebra 2 5-2 Properties of Quadratic Functions in Standard Form Lesson Quiz: Part II Consider the function f(x)= 2x 2 + 6x – 7. 6. Graph the function. 7. Find the domain and range of the function.