Determine the following algebraically (no calculator) a)Vertex b)x-intercepts c)y- intercepts d)Is the vertex a max or min? How do you know without graphing? Determine the following algebraically (no calculator) a)Vertex b)x-intercepts c)y- intercepts d)Is the vertex a max or min? How do you know without graphing? Quadratics

Section 3.2 Polynomial Functions and Models

1. Polynomial Functions and their degree Find the degree (largest power when written in standard form). Factored form: Add the degrees contributed by each factor A polynomial function is a function of the form Coefficients are: n is non-negative integer.

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: Symmetry: Domain: Range:

3. End Behavior of Polynomials Given a polynomial As x increases/decreases without bound, the highest degree term determines the end behavior of f(x).

3. End Behavior of the graph Even Degree and Even Degree and Leading Term Test look at leading term (from standard form) Leading Coefficient is Positive Leading coeffic. is Negative Leading coeffic. is Negative Odd Degree and Odd Degree and Leading coefficient is Positive Leading coefficient is Negative

3b) Determine End Behavior using Leading Coefficient Test 1) 2) 3) 1) 2) 3) p. 183 # What is the end behavior of these graphs?

3c) End Behavior can also be used to determine Window Size on Calculator How do you know if you are viewing the entire graph? Set window to [-5,5] x [-5,5] and graph:

4. Zeroes of Polynomials and their Multiplicity Factor Theorem If is a polynomial function, for which the following are equivalent statements : (1) r is a zero or root of (2) r is an x-intercept of (3) is a factor of Example 1: Determine all zeros of f.

4 Zeroes of Polynomials and their Multiplicity Definition: The multiplicity of a zero is the degree of the factor Example: If f has a factor, then 3 is a zero with multiplicity 4 Example: Determine: The zeros of f and multiplicity of each : Example: Determine: The zeros of f and multiplicity of each : Multiplicity greater than 1 represents a repeated zero

5. Investigating the Role of Multiplicity. How does Multiplicity affect the behavior of the graph at the zero? Graph each function, what do you notice changes at the zero when the degree is even or odd? 1) 3) 2) 4) Does the sign of change on each side of the zero?

6. Behavior of Graph at a Zero Multiplicity tells us the behavior of the graph at a zero (x- intercept). If multiplicity, m, is a number that is: Behavior of graph at the zero Eventouches (is tangent) Oddcrosses (changes sign) The graph flattens out at the zero as the multiplicity increases.

6. Example Continued ZerosMultiplicityIs m an even or odd integer? Behavior of graph at the zero Previous Example.

6. More Practice State the degree of this polynomial. How many zeros does this function have? Zeros of the function Multiplicity of the zero Is m even or odd? Shape of the graph near the zero

7. Graphing a Polynomial Function Given the polynomial 1.Leading Term and End behavior: 2.y-intercept: 3.x-intercepts (i.e. the zeros) ZerosMultiplicityeven or odd?Cross/Touch HINT HINT: Graph the end behavior at outermost x-intercepts first. Fill in the rest according to the table. HINT HINT: Graph the end behavior at outermost x-intercepts first. Fill in the rest according to the table.

7. Graphing a Polynomial Function Before finding zeros: Write in completely factored form ZerosMultiplicityeven or odd?Cross/Touch 1.Leading Term and End behavior: 2.y-intercept: 3.x-intercepts

7. Graphing a Polynomial Function ZerosMultiplicityeven or odd?Cross/Touch 1.Leading Term and End behavior: 2.y-intercept: 3.x-intercepts

1.End behavior 2.y-intercept 3.x-intercepts (i.e. the zeros) 7. Graphing a Polynomial Function When polynomial is not in factored form, find zeros using: Factoring or Graphing Calculator. When polynomial is not in factored form, find zeros using: Factoring or Graphing Calculator. ZerosMultiplicityeven or odd?Cross/Touch

10. Using the graphing calculator Graph: p. 184 #81. x-intercepts: Use ZERO feature y-intercepts: TRACE: x=0 d) Table to determine graph close to zero. Is it above or below? e) Max/Min Find zeros (x-intercepts) using graphing calculator.

Recall…. Factor Theorem If is a polynomial function, for which the following are equivalent statements : (1) r is a zero or root of (2) r is an x-intercept of (3) is a factor of

9. a) Building Polynomials from zeros Given that f is a polynomial with zeros at each with multiplicity we can write: Write one possible polynomial with these properties: 1)Zeroes at: -1, 2, 5 ; Each with multiplicity 1 2) x-intercepts at: (-3,0), (4,0), (-1,0), (0,0) and the graph rises to the left and falls to the right.

Polynomial – turning points Definition: turning point - graph changes between increasing and decreasing. n If polynomial is degree n n-1 Then it will have AT MOST n-1 turning points.

9. a)Building Polynomials from a graph Use y-intercept to find scale factor. Zeros x-intercepts C/T ?Multiplicity

More practice…. Construct a polynomial function that might have this graph. Use higher-degree if graph is “flat” at the zero Zeros x-intercepts C/T ?Multiplicity

10. Finding a function of best fit Using the graphing calculator, we can find a function of best fit for the following relationships: Quadratic : Cubic: Count the # of turning points to determine best function Count the # of turning points to determine best function

11. More practice

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