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Mastering Polynomial Functions: Real Zeros, Graphs, and End Behaviors

Learn about polynomial functions, identifying real zeros and their multiplicities, graph properties, and end behaviors. Practice finding, graphing, and analyzing polynomial functions.

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Mastering Polynomial Functions: Real Zeros, Graphs, and End Behaviors

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  1. Do Now: Solve the inequality

  2. Academy Algebra II/Trig 5.1: Polynomial Functions and Models HW: p.340 (12, 13, 17-20, 40, 41, 43, 45-47 – parts a,d,e only) Test 4.3-4.5, 5.1, 5.5-5.6:

  3. Vocabulary • Polynomial Function = a function in the form: where , exponents are whole #’s, and coefficients are real. • Standard Form = terms are written in descending order of exponents. • Degree = the highest exponent.

  4. Common Polynomial Functions

  5. Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.

  6. Identify the real zeros of a polynomial function and their multiplicity. • If a polynomial is factored completely, it is easy to solve the question f(x) = 0 using the zero-product property. Example: Find the real zeros of the function:

  7. Real Zeros • For the polynomial: • 7 is a zero of multiplicity 1 because the exponent on the factor of x – 7 is 1, • -3 is a zero of multiplicity 2 because the exponent on the factor of x + 3 is 2. List each real zero and its multiplicity.

  8. Form a polynomial whose real zeros and degree are given. 1.) Zeros: -3, 0, 4; degree 3 2.) Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3

  9. Graphs of a polynomial function • Graphs are smooth (no corners) and continuous (no breaks). • Determine which graphs are not polynomials.

  10. Graphs of a polynomial function – turning points. • If f is a polynomial function of degree n, then f has at most n – 1 turning points.

  11. End Behaviors of a Polynomial Function • Degree: Even • Leading Coefficient: Negative • Degree: Even • Leading Coefficient: Positive

  12. End Behaviors of a Polynomial Function Degree: Odd Leading Coefficient: Positive • Degree: Odd • Leading Coefficient: Negative

  13. Do Now: Which of the graphs could be f(x) = x4 + 5x3 + 5x2 – 5x – 3? • Hint: Identify the y-intercept to help eliminate options.

  14. Academy Algebra II/Trig 5.1: Finish HW: p.341-342 (57-60 all; 65,70,74 – part f: like class work)

  15. Graph the polynomial. Label intercepts, determine turning points, and end behavior. (May use graphing calculator for shape between intercepts.)

  16. Graph the polynomial. Label intercepts, determine turning points, and end behavior. (May use graphing calculator for shape between intercepts.)

  17. Graph the polynomial. Label intercepts, determine turning points, and end behavior. (May use graphing calculator for shape between intercepts.)

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