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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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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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f (x)
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f (x)
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-2
-2
-4
2
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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PDF Pass
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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2-2-4
24
16
8
f (x)
← 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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