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Chapter 5 23 Glencoe Algebra 2

Graphs of Polynomial Functions

Determine consecutive integer values of x between which each real zero of f(x) = 2x4 - x3 - 5 is located. Then draw the graph.

Make a table of values. Look at the values of f(x) to locate the zeros. Then use the points to sketch a graph of the function. The changes in sign indicate that there are zeros between x = -2 and x = -1 and between x = 1 and x = 2.

ExercisesGraph each function by making a table of values. Determine the values of x between which each real zero is located.

1. f (x) = x3 - 2x2 + 1 2. f (x) = x4 + 2x3 - 5 3. f(x) = -x4 + 2x2 - 1

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f (x)

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f (x)

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f (x)

4. f (x) = x3 - 3x2 + 4 5. f (x) = 3x3 + 2x - 1 6. f(x) = x4 - 3x3 + 1

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f (x)

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-2

-2

-4

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42

Location PrincipleSuppose y = f (x) represents a polynomial function and a and b are two numbers such that

f (a) < 0 and f (b) > 0. Then the function has at least one real zero between a and b.

x f(x)

-2 35

-1 -2

0 -5

1 -4

2 19

Study Guide and InterventionAnalyzing Graphs of Polynomial Functions

Example

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Chapter 5 24 Glencoe Algebra 2

Maximum and Minimum Points A quadratic function has either a maximum or a minimum point on its graph. For higher degree polynomial functions, you can find turning points, which represent relative maximum or relative minimum points.

Graph f(x) = x3 + 6x2 - 3. Estimate the x-coordinates at which the relative maxima and minima occur.Make a table of values and graph the function.

ExercisesGraph each polynomial function. Estimate the x-coordinates at which the relative maxima and relative minima occur.

1. f (x) = x3 - 3x2 2. f (x) = 2x3 + x2 - 3x 3. f (x) = 2x3 - 3x + 2

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f (x)

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f (x)

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4. f (x) = x4 - 7x - 3 5. f (x) = x5 - 2x2 + 2 6. f (x) = x3 + 2x2 - 3

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← indicates a relative maximum

← zero between x = -1, x = 0

← indicates a relative minimum

x f (x)

-5 22

-4 29

-3 24

-2 13

-1 2

0 -3

1 4

2 29

Study Guide and Intervention (continued)

Analyzing Graphs of Polynomial Functions

Example

A relative maximum occurs at x = -4 and a relativeminimum occurs at x = 0.

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