1. 1 of 18 Pre-Cal Chapter 2 Section 1 SAT/ACT Warm - up

2. 2 of 18 Pre-Cal Chapter 2 Section 1 Chapter 2 – Vocabulary Words 1.Polynomial function 2.Linear function 3.Quadratic function 4.Continuous 5.Intermediate Value Theorem 6.Synthetic division 7.Imaginary number 8.Complex conjugates 9.Rational function 10.Slant Asymptote See Page 172

3. Graphing Quadratic Functions Chapter2 Sec 1 Chapter 2 Sec 1

4. 4 of 18 Pre-Cal Chapter 2 Section 1 Essential Question How do you sketch graphs and write equations of parabolas? Key Vocabulary: Polynomial Function Linear equation Quadratic function

5. 5 of 18 Pre-Cal Chapter 2 Section 1Definitions

6. 6 of 18 Pre-Cal Chapter 2 Section 1 Quadratic function The graph of a quadratic function is a parabola. Every parabola is symmetrical about a line called the axis (of symmetry). The intersection point of the parabola and the axis is called the vertex of the parabola. x y axis f (x) = ax 2 + bx + c vertex

7. 7 of 18 Pre-Cal Chapter 2 Section 1

8. 8 of 18 Pre-Cal Chapter 2 Section 1 Leading Coefficient The leading coefficient of ax 2 + bx + c is a. When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum. When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum. x y f(x) = ax 2 + bx + c a > 0 opens upward vertex minimum x y f(x) = ax 2 + bx + c a < 0 opens downward vertex maximum

9. 9 of 18 Pre-Cal Chapter 2 Section 1 5 y x 5-5-5 Simple Quadratic Functions The simplest quadratic functions are of the form f (x) = ax 2 (a  0) These are most easily graphed by comparing them with the graph of y = x 2. Example: Compare the graphs of

10. 10 of 18 Pre-Cal Chapter 2 Section 1 Example: f(x) = (x –3) 2 + 2 Example: Graph f (x) = (x – 3) 2 + 2 and find the vertex and axis. f (x) = (x – 3) 2 + 2 is the same shape as the graph of g (x) = (x – 3) 2 shifted upwards two units. g (x) = (x – 3) 2 is the same shape as y = x 2 shifted to the right three units. f (x) = (x – 3) 2 + 2 g (x) = (x – 3) 2 y = x 2 - 4- 4 x y 4 4 vertex (3, 2)

11. 11 of 18 Pre-Cal Chapter 2 Section 1 Completing the Square 1.Group together the x’s 2.Factor out the a. 3.Take coefficient of x. a.Divide by 2 b.Then square result 4.Add the “squared result” inside the parenthesis and subtract the “Squared result times a” outside the parenthesis. 5. Factor the parenthesis, it will be (x ± (3a.)) 2.

12. 12 of 18 Pre-Cal Chapter 2 Section 1 x y Quadratic Function in Standard Form Example: Graph the parabola f (x) = 2x 2 + 8x + 5 and find the axis and vertex. f (x) = 2x 2 + 8x + 5 original equation f (x) = 2( x 2 + 4x) + 5 group the x’s & factor out 2 f (x) = 2( x 2 + 4x + 4) + 5 – 8 complete the square f (x) = 2( x + 2) 2 – 3 standard form a > 0  parabola opens upward like y = 2x 2. h = –2, k = –3  axis x = –2, vertex (–2, –3). x = –2 f (x) = 2x 2 + 8x + 5 (–2, –3)

13. 13 of 18 Pre-Cal Chapter 2 Section 1 x y 4 4 Vertex and x-Intercepts Example: Graph and find the vertex and x-intercepts of f (x) = –x 2 + 6x + 7. f (x) = – x 2 + 6x + 7 original equation f (x) = – ( x 2 – 6x) + 7 factor out –1 f (x) = – ( x 2 – 6x + 9) + 7 + 9 complete the square f (x) = – ( x – 3) 2 + 16 standard form a < 0  parabola opens downward. h = 3, k = 16  axis x = 3, vertex (3, 16). Find the x-intercepts by solving –x 2 + 6x + 7 = 0. (–x + 7 )( x + 1) = 0 factor x = 7, x = –1 x-intercepts (7, 0), (–1, 0) x = 3 f(x) = –x 2 + 6x + 7 (7, 0)(–1, 0) (3, 16)

14. 14 of 18 Pre-Cal Chapter 2 Section 1 y x f (x) = a(x – h) 2 + k standard form f (x) = a(x – 2) 2 + (–1)vertex (2, –1) = (h, k) y = f(x) (0, 1) (2, –1) Since (0, 1) is a point on the parabola: f (0) = a(0 – 2) 2 – 1 1 = 4a –1 and Example: Find an equation for the parabola with vertex (2, –1) passing through the point (0, 1).

15. 15 of 18 Pre-Cal Chapter 2 Section 1 Example: Find the vertex of the graph of f (x) = x 2 – 10x + 22. f (x) = x 2 – 10x + 22 original equation a = 1, b = –10, c = 22 At the vertex, So, the vertex is (5, -3). The vertex of the graph of f (x) = ax 2 + bx + c (a  0) Vertex of a Parabola

16. 16 of 18 Pre-Cal Chapter 2 Section 1 Minimum and Maximum Values of Quadratic Functions

17. 17 of 18 Pre-Cal Chapter 2 Section 1 The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet. Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:

18. 18 of 18 Pre-Cal Chapter 2 Section 1 Essential Question How do you sketch graphs and write equations of parabolas?

19. 19 of 18 Pre-Cal Chapter 2 Section 1 Daily Assignment Chapter 2 Section 1/ Ch 1 Review Text Book (TB) Textbook Pgs 99– 102 #1 – 4, 7, 11, 17, 21, 27, 29, 31, 35, 55, 63 – 67 All R Read Sections 2.2 – 2.3 Show all work for credit.