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Math  /  Algebra

QuestionPart 7 of 7 HW Score: 39.75%,32.239.75 \%, 32.2 of 81 points Points: 0 of 1 Save
For the polynomial function below. (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the xx-axis at each xx-intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of ff resembles for large values of ∣x∣|x|. f(x)=−2(x−9)(x+9)2f(x)=-2(x-9)(x+9)^{2} the mulupicity or the splanet cero is <. (Type a whole number.) (b) The graph crosses the xx-axis at the larger xx-intercept.
The graph touches the xx-axis at the smaller xx-intercept. (c) The maximum number of turning points on the graph is 2 . (Type a whole number.) (d) Type the power function that the graph of f resembles for large values of ∣x∣|\mathrm{x}|. y=□\mathrm{y}=\square

Studdy Solution

STEP 1

1. We are given a polynomial function f(x)=−2(x−9)(x+9)2 f(x) = -2(x-9)(x+9)^2 .
2. We need to analyze the polynomial to find real zeros, their multiplicities, the behavior at each zero, the maximum number of turning points, and the end behavior.

STEP 2

1. Identify and list each real zero and its multiplicity.
2. Determine whether the graph crosses or touches the x-axis at each x-intercept.
3. Determine the maximum number of turning points on the graph.
4. Determine the end behavior of the polynomial function.

STEP 3

Identify and list each real zero and its multiplicity:
The polynomial is given as f(x)=−2(x−9)(x+9)2 f(x) = -2(x-9)(x+9)^2 .
- The factor (x−9) (x-9) gives a zero at x=9 x = 9 with multiplicity 1. - The factor (x+9)2 (x+9)^2 gives a zero at x=−9 x = -9 with multiplicity 2.

STEP 4

Determine whether the graph crosses or touches the x-axis at each x-intercept:
- For x=9 x = 9 (multiplicity 1), the graph crosses the x-axis. - For x=−9 x = -9 (multiplicity 2), the graph touches the x-axis and does not cross.

STEP 5

Determine the maximum number of turning points on the graph:
The degree of the polynomial is the sum of the multiplicities, which is 1+2=3 1 + 2 = 3 .
The maximum number of turning points is one less than the degree of the polynomial:
Maximum turning points=3−1=2 \text{Maximum turning points} = 3 - 1 = 2

STEP 6

Determine the end behavior of the polynomial function:
The leading term of the polynomial, when expanded, is determined by the highest power of x x from each factor:
f(x)=−2(x)(x2)=−2x3 f(x) = -2(x)(x^2) = -2x^3
For large values of ∣x∣ |x| , the graph of f f resembles the power function:
y=−2x3 y = -2x^3
The real zeros are x=9 x = 9 (multiplicity 1) and x=−9 x = -9 (multiplicity 2). The graph crosses the x-axis at x=9 x = 9 and touches at x=−9 x = -9 . The maximum number of turning points is 2. The end behavior resembles the power function y=−2x3 y = -2x^3 .

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