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3.7 Investigating Graphs of Polynomial Functions

3.7 Investigating Graphs of Polynomial Functions. Objective : Students will be able to use properties of end behavior to analyze and describe graphs of polynomial functions. . Graphs of Polynomial Functions . Table on pg. 197 Vocabulary:

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3.7 Investigating Graphs of Polynomial Functions

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  1. 3.7 Investigating Graphs of Polynomial Functions Objective: Students will be able to use properties of end behavior to analyze and describe graphs of polynomial functions.

  2. Graphs of Polynomial Functions • Table on pg. 197 • Vocabulary: • End behavior: a description of the values of the function as x approaches positive infinity or negative infinity. It is determined by the degree and leading coefficient of the polynomial function.

  3. Polynomial End Behavior

  4. Determining End Behavior of Polynomial Functions • Identify the leading coefficient, degree, and end behavior. • Example 1: P(x) = 2x5 + 3x2 – 4x – 1 • Leading Coefficient: • Degree: • End Behavior:

  5. Example 2 • P(x) = -3x2 + x + 1 • Leading Coefficient: • Degree: • End Behavior:

  6. Example 3 – you try • P(x) = 4x4 + 3x3 – 2x2 + x – 3 • Leading Coefficient: • Degree: • End Behavior:

  7. Using Graphs to Analyze Polynomial Functions • Use the end behavior to identify whether the function graphed has an odd or even degree and a positive or negative coefficient. • Example 4:

  8. Example 5

  9. Example 6 – you try

  10. Homework for tonight • Textbook pg. 201 # 15, 16, 19 – 22

  11. Writing Polynomial Equations with multiplicity • Solving a polynomial equation means to find the roots of the equation. The multiplicity of each root determines whether the graph of the equation crosses the x-axis or just touches the x-axis. • Also, depending on the degree of the polynomial the graph takes on a different shape and begins at a different point. (chart below) • Even multiplicity – curve touches the x-axis • Odd multiplicity – line passes through the x-axis

  12. For example… • The graph at right below shows x-intercepts of 5, 0, 2, and 6. The curve just touches the x-axis at 2, indicating an even multiplicity, and passes through the x-axis at 5, 0, and 6, indicating an odd multiplicity. • An equation for this polynomial function is y(x 5)(x)(x 2) 2(x 6). • This is not the only equation for this graph, just one in the “family of functions”.

  13. Example 7 • Directions: Write a possible polynomial function whose graph fits the given conditions. • Crosses the x-axis at -3, 0, and 4; lies above the x-axis between -3 and 0 and lies below the x-axis between 0 and 4.

  14. Example 8 • Touches the x-axis at -1; crosses the x-axis at 1 and 3, and lies above the x-axis between 1 and 3.

  15. Example 9 • Has x-intercepts of -5, -2, ½, and 3; lies above the x-axis between -5 and -2 and also between -2 and ½

  16. Example 10 … your turn • Has x-intercepts of -6, 0, 1, 3, and 4; lies above the x-axis at all points between the x-intercepts except on the interval from 0 to 1.

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