Chapter 3: Polynomial Functions

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. 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. 3.2 Completing the Square • Completing The Square • Divide both sides of the equation by a, so that the coefficient of is 1. • Add to both sides. • Add to both sides the square of half the coefficient of x, • Factor the right side as the square of a binomial and combine terms on the left. • Isolate the term involving P(x) on the left. • Multiply both sides by a. • Rewrite in the form

5. 3.2 Example of Completing the Square Divide by 2 to make the coefficient of x2equal 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. 3.2 Example of Completing the Square • From we can determine several components of the graph of

7. 3.2 Graphs of Quadratic Functions Transform into • P has vertex (-3,1), so the • graph of f(x) = x2 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. 3.2 Graph of P(x) = a(x-h)2 + k • The graph of • is a parabola with vertex (h,k), and the vertical line • x = h as axis of symmetry; • opens upward if a > 0 and downward if a < 0; • is broader than and narrower than • 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.

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. 3.2 Vertex Formula The vertex of the graph of is the point 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

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 • if a > 0, the vertex (h,k) is called the minimum point of the graph. The minimumvalue of the function is P(h) = k. • 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. 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. 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. 3.2 Applications and Modeling 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 • 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.

15. 3.2 Height of a Propelled Object 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 s0 feet with initial velocity v0 feet per second is where t is the number of seconds after the object is propelled. • The coefficient of t², 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.

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

17. 3.2 Solving a Problem Involving Projectile Motion (d) After how many seconds does the projectile reach its maximum height? • 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. 3.2 Solving a Problem Involving Projectile Motion Figure 21 pg 3-25 (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.