Graphing a Quadratic

1. Graphing a Quadratic Warm-up • Determine the following algebraically (no calculator) • vertex • x- and y-intercepts. • Is the vertex a max or min? How would you know without graphing? • Identify intervals of increasing/decreasing. • Write f in standard form (complete the square)

2. Section 3.2 Polynomial Functions and Models • Polynomial Functions and their Degree • Properties of Power Functions • Zeros of a Polynomial and Multiplicity • Building Polynomials • End Behavior of Polynomials • Leading Coefficient Test • Analyzing the Graph of a Polynomial Function

3. 1. Polynomial Functions and their degree A polynomial function is a function of the form where the coefficients are real numbers and n is a nonnegative integer. The degree of the function is the largest power of x. Factored form: Add the degrees

4. Polynomial Functions and their degree Determine which of the following are polynomial functions. For those that are, state the degree. 1. 2. 3. 4. 5.

5. Polynomial Functions Determine which of the following are polynomial functions. For those that are, state the degree. 6. 7. 8. 9. 10.

6. Polynomial Functions and their degree Determine which of the following are polynomial functions. For those that are, state the degree. T; 5 6. T; 3 2. F 7. T; 7 3. T; 5 8. T; 3 4. F 9. F 5. T; 0 10. F

7. 2. Properties of the Power Functions A power function is of the form If n is odd integer If n is even integer Symmetry: Domain: Range: Key Points:

8. 3. Zeroes of Polynomials and their Multiplicity • Factor TheoremIf is a polynomial function, the following are equivalent statements: • r is a real number for which: • r is called a zero or root of • r is an x-intercept of the graph of • is a factor of Example: Factor For each factor, summarize properties 1-4 of the Factor theorem..

9. 3. Zeroes of Polynomials and their Multiplicity Definition: The multiplicity of a zero is the degree of the factor Notation: If f has a factor we say is a zero of multiplicity Example: Identify the zeros and their multiplicities of: Degree of f = Graph f(x).

10. 3. Zeroes of Polynomials and their Multiplicity Example – continued. What does the value of m tell us about the graph ?

11. 3. a) Large values of Multiplicity Analyze the graph of What happens at the zero as m gets large? Use your graphing calculator to graph the following: 1) 2) The graph flattens out at the zero as the multiplicity increases.

12. 3. More Practice State the degree of this polynomial. How many zeros does this function have?

13. 4. Building Polynomials Given that f has zeros with multiplicity we can write: Write a polynomial with these properties: 1) Degree 4: Zeroes at: -5, -4, 0, 2, (multiplicity 1) 2) Degree 3: Zeroes at: -3, multiplicity 2; 5, multiplicity 1 and passing through the point (0,9)

14. 4. Building Polynomials Construct a polynomial function that might have this graph.

15. 5. End Behavior of Polynomials The end behavior of the graph is determined by the coefficient and degree of highest degree term Sketch the graph of these functions. What is the end behavior?

16. 5. a) Leading Coefficient Test Leading term determines end behavior 1. For n even. 2. For n odd. Rises to left and falls to right Rises to left and rises to right Falls to left and falls to right Falls to left and rises to right

17. 6. Analyzing the Graph of a Polynomial Function Given the polynomial What is the degree of this polynomial? 1. End behavior. 2. x-intercepts. Solve f(x) = 0 Behavior at each intercept (even/odd) b) If k > 1, graph flattens for larger values of k. 3. y-intercepts. Find f(0). 4. Symmetry: Odd/Even 5. Turning points: Graph changes between increasing/decreasing.

18. Zeroes of Polynomials