1. Copyright © 2007 Pearson Education, Inc. Slide 3-1

2. Copyright © 2007 Pearson Education, Inc. Slide 3-2 Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

3. Copyright © 2007 Pearson Education, Inc. Slide 3-3 3.2 Quadratic Functions and Graphs Quadratic Functions are polynomial functions, discussed later. P is often used to represent a polynomial function. A function of the form with a  0 is called a quadratic function. Recall is the graph of stretched or shrunk and shifted horizontally and vertically. Example Figure 9 pg 3-13

4. Copyright © 2007 Pearson Education, Inc. Slide 3-4 3.2 Completing the Square Rewrite in the form Completing The Square 1.Divide both sides of the equation by a, so that the coefficient of is 1. 2.Add to both sides. 3.Add to both sides the square of half the coefficient of x, 4.Factor the right side as the square of a binomial and combine terms on the left. 5.Isolate the term involving P(x) on the left. 6.Multiply both sides by a.

5. Copyright © 2007 Pearson Education, Inc. Slide 3-5 3.2 Example of Completing the Square Divide by 2 to make the coefficient of x 2 equal to 1. Add 8 to both sides. Add [½·2] 2 to both sides to complete the square on the right. Combine terms on the left; factor on the right. Subtract 9 from both sides. Multiply both sides by 2.

6. Copyright © 2007 Pearson Education, Inc. Slide 3-6 3.2 Example of Completing the Square From we can determine several components of the graph of

7. Copyright © 2007 Pearson Education, Inc. Slide 3-7 3.2 Graphs of Quadratic Functions Transform into - P has vertex (-3,1), so the graph of f (x) = x 2 is shifted left 3 and up 1. - The coefficient of (x+3) 2 is –1, so the graph opens downward. - y-intercept: (0,–8) - Axis of symmetry: line x = -3 - Domain: (- ,  ); Range: (- ,1] - increasing: (- ,-3]; decreasing: [-3,  )

8. Copyright © 2007 Pearson Education, Inc. Slide 3-8 One method to determine the coordinates of the vertex is to complete the square. Rather than go through the procedure for each individual function, we generalize the result for P(x) = ax² + bx + c. 3.2 Graph of P(x) = a(x-h) 2 + k The graph of (a)is a parabola with vertex (h,k), and the vertical line x = h as axis of symmetry; (a)opens upward if a > 0 and downward if a < 0; (b)is broader than and narrower than

9. Copyright © 2007 Pearson Education, Inc. Slide 3-9 3.2 Vertex Formula for Parabola P(x) = ax² + bx + c (a  0) Standard form Replace P(x) with y to simplify notation. Divide by a. Subtract Add Combine terms on the left; factor on the right. Get y-term alone on the left. Multiply by a and write in the form

10. Copyright © 2007 Pearson Education, Inc. Slide 3-10 3.2 Vertex Formula Example Use the vertex formula to find the coordinates of the vertex of the graph of Analytic Solution – exact solution Approximation Using a calculator, we find The vertex of the graph of is the point

11. Copyright © 2007 Pearson Education, Inc. Slide 3-11 3.2 Extreme Values The vertex of the graph of is the –lowest point on the graph if a > 0, or –highest point on the graph if a < 0. Such points are called extreme points (also extrema, singular: extremum). For the quadratic function defined by (a)if a > 0, the vertex (h,k) is called the minimum point of the graph. The minimum value of the function is P(h) = k. (b)if a < 0, the vertex (h,k) is called the maximum point of the graph. The maximum value of the function is P(h) = k.

12. Copyright © 2007 Pearson Education, Inc. Slide 3-12 3.2 Identifying Extreme Points and Extreme Values Example Give the coordinates of the extreme point and the corresponding maximum or minimum value for each function. (a)(b) The vertex of the graph is (–1,–18). Since a > 0, the minimum point is (–1,–18), and the minimum value is –18. The vertex of the graph is (–3,1). Since a < 0, the maximum point is (–3,1), and the maximum value is 1.

13. Copyright © 2007 Pearson Education, Inc. Slide 3-13 3.2 Finding Extrema with the Graphing Calculator Let One technique is to use the fmin function. We get the x-value where the minimum occurs. The y-value is found by substitution. Figure 14 pg 3-20b

14. Copyright © 2007 Pearson Education, Inc. Slide 3-14 Example The table gives data for the percent increase (y) on hospital services in the years 1994 – 2001, where x is the number of years since 1990. The data are plotted in the scatter diagram. A good model for the data is the function defined by (a)Use f (x) to approximate the year when the percent increase was a minimum. The x-value of the minimum point is (b)Find the minimum percent increase. The minimum value is.64 differing slightly from the data value of.5 in the table. 3.2 Applications and Modeling Year xPercent increase y YearPercent increase y 41.883.4 50.895.8 60.5107.1 71.31112.0

15. Copyright © 2007 Pearson Education, Inc. Slide 3-15 3.2 Height of a Propelled Object The coefficient of t ²,  16, is a constant based on gravitational force and thus varies on different surfaces. Note that s(t) is a parabola, and the variable x will be used for time t in graphing-calculator-assisted problems. Height of a Propelled Object If air resistance is neglected, the height s (in feet) of an object propelled directly upward from an initial height s 0 feet with initial velocity v 0 feet per second is where t is the number of seconds after the object is propelled.

16. Copyright © 2007 Pearson Education, Inc. Slide 3-16 A ball is thrown directly upward from an initial height of 100 feet with an initial velocity of 80 feet per second. (a)Give the function that describes height in terms of time t. (b)Graph this function. (c)The cursor in part (b) is at the point (4.8,115.36). What does this mean? 3.2 Solving a Problem Involving Projectile Motion After 4.8 seconds, the object will be at a height of 115.36 feet.

17. Copyright © 2007 Pearson Education, Inc. Slide 3-17 3.2 Solving a Problem Involving Projectile Motion (d) After how many seconds does the projectile reach its maximum height? (e)For what interval of time is the height of the ball greater than 160 feet? Figure 19 pg 3-24 Using the graphs, t must be between.92 and 4.08 seconds.

18. Copyright © 2007 Pearson Education, Inc. Slide 3-18 3.2 Solving a Problem Involving Projectile Motion (f) After how many seconds will the ball fall to the ground? When the ball hits the ground, its height will be 0, so we need to find the positive x-intercept. From the graph, the x-intercept is about 6.04, so the ball will reach the ground 6.04 seconds after it is projected. Figure 21 pg 3-25