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Warm Up

Understand how to analyze polynomial functions, find real solutions, and identify key points on graphs to determine turning points and zeros. Learn about factors, solutions, and intercepts in polynomial equations.

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Warm Up

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  1. Warm Up • What do you know about the graph of f(x) = (x – 2)(x – 4)2 ?

  2. 5.4 Analyzing Graphs of Polynomial Functions

  3. Zeros, Factors, Solutions, and Intercepts Let f(x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0be a polynomial function. The following statements are equivalent. • Zero:k is a zero of the polynomial function f. • Factor: x – k is a factor of the polynomial f(x). • Solution:k is a solution of the polynomial equation f(x) = 0. • x-intercept:k (if it is real) is an x-intercept of the graph of the polynomial function f.

  4. Find approximate real solutions for polynomial equations by graphing 1. Put polynomial equation into standard form. 2. Set equal to y 3. Graph 4. Find the zeros/ real solutions table – y-value of 0 calc – zero

  5. Example 1: • x3 + 6x2 – 4x – 24 = 0 b. x4 – 9x2 = 0

  6. Example 2 Determine consecutive values of x between whicheach real zero of the function f(x) = x4 – x3 – 4x2 + 1 is located using the table below.

  7. To identify key points on a graph of a polynomial function • Turning points are called local maximums and local minimums because they are the highs or lows for the area. • If there are two turning points without a zero between them  there are imaginary zeros.

  8. Example 3: • Estimate the coordinates of each turning point and state whether each corresponds to a local max or local min. Then list all real zeros and determine the least degree the function can have.

  9. Example 4 Graph each function. Identify the x-intercepts (zeros), local maximums, and local minimums. a. f(x) = x3 + 2x2 – 5x + 1

  10. b. f(x) = 2x4 – 5x3 – 4x2 – 6

  11. Example 5 • The weight w, in pounds, of a patient during a 7-week illness is modeled by the function • w(n) = 0.1n3 – 0.6n2 + 110, where n is the number of weeks since the patient became ill. • Graph the equation by making a table of values for weeks 0 through 7. Plot the points and connect with a smooth curve. • Describe the turning points of the graph and its end behavior.

  12. There is a relative minimum atweek 4. w(n) → ∞ asn→ ∞.

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