Math / Algebra

QuestionAnalyze the polynomial function $f(x)=-4(x+4)(x-3)^{3}$ using parts (a) through (h) below. (a) Determine the end behavior of the graph of the function.
The graph of $f$ behaves like $y=$ $\square$ for large values of $|x|$ . (b) Find the $x$ -and $y$ -intercepts of the graph of the function.
The $x$ -intercept(s) is/are $\square$ . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
The $y$ -intercept(s) is/are $\square$ 1. (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) (c) Determine the real zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
The real zero(s) of $f$ is/are $\square$ (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)
The lesser zero is a zero of multiplicity $\square$ , so the graph of $f(1)$ $\qquad$ the $x$ -axis at $x=$ $\qquad$ . The greater zero is a zero of multiplicity $\square$ so the graph off (2) $\qquad$ the $x$ -axis at $x=$ $\square$ . (d) Use a graphing utility to graph the function. The graphs are shown in the viewing window $X_{\min }=-10, X_{\max }=10, X_{s d l}=1$ , $Y_{\min }=-2100, Y_{\max }=2100 Y_{\text {scl }}=210$ . Choose the correct graph below. A. B. c. D.

Studdy Solution

STEP 1

1. The polynomial function is given by $f(x) = -4(x+4)(x-3)^3$ .
2. We need to analyze the polynomial based on its end behavior, intercepts, zeros, and graph.

STEP 2

1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts and $y$ -intercepts.
3. Determine the real zeros and their multiplicities.
4. Use a graphing utility to visualize the function.

STEP 3

To determine the end behavior of the polynomial function $f(x) = -4(x+4)(x-3)^3$ , we focus on the leading term when the polynomial is expanded. The leading term will dominate the behavior for large values of $|x|$ .
The polynomial can be expanded to its leading term by considering the highest power of $x$ from each factor:
$f(x) = -4(x)(x^3) = -4x^4$
The leading term is $-4x^4$ .

STEP 4

The end behavior of the polynomial is determined by the leading term $-4x^4$ :
- As $x \to \infty$ , $-4x^4 \to -\infty$ . - As $x \to -\infty$ , $-4x^4 \to -\infty$ .
Thus, the graph of $f$ behaves like $y = -4x^4$ for large values of $|x|$ .

STEP 5

Find the $x$ -intercepts by setting $f(x) = 0$ :
$-4(x+4)(x-3)^3 = 0$
The solutions are:
$x+4 = 0 \quad \Rightarrow \quad x = -4$ $(x-3)^3 = 0 \quad \Rightarrow \quad x = 3$
Thus, the $x$ -intercepts are $x = -4$ and $x = 3$ .

STEP 6

Find the $y$ -intercept by evaluating $f(0)$ :
$f(0) = -4(0+4)(0-3)^3$ $f(0) = -4(4)(-27)$ $f(0) = 432$
The $y$ -intercept is $y = 432$ .

STEP 7

Determine the real zeros and their multiplicities:
- $x = -4$ is a zero of multiplicity 1 (since it appears once in the factor $(x+4)$ ). - $x = 3$ is a zero of multiplicity 3 (since it appears three times in the factor $(x-3)^3$ ).

STEP 8

Based on multiplicity:
- The graph crosses the $x$ -axis at $x = -4$ because the multiplicity is odd. - The graph touches the $x$ -axis at $x = 3$ because the multiplicity is odd, but since it's a cubic term, it will also cross.

STEP 9

Use a graphing utility to graph the function. Given the viewing window and scale, choose the correct graph that matches the end behavior, intercepts, and zero behavior.
The analysis of the polynomial function is complete. The graph of $f$ behaves like $y = -4x^4$ for large values of $|x|$ , has $x$ -intercepts at $x = -4$ and $x = 3$ , a $y$ -intercept at $y = 432$ , and crosses the $x$ -axis at both intercepts due to their multiplicities. Use a graphing utility to confirm these features.

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Studdy Solution

STEP 1

1. The polynomial function is given by $f(x) = -4(x+4)(x-3)^3$ .
2. We need to analyze the polynomial based on its end behavior, intercepts, zeros, and graph.

STEP 2

1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts and $y$ -intercepts.
3. Determine the real zeros and their multiplicities.
4. Use a graphing utility to visualize the function.

STEP 3

STEP 4

The end behavior of the polynomial is determined by the leading term $-4x^4$ :
- As $x \to \infty$ , $-4x^4 \to -\infty$ . - As $x \to -\infty$ , $-4x^4 \to -\infty$ .
Thus, the graph of $f$ behaves like $y = -4x^4$ for large values of $|x|$ .

STEP 5

Find the $x$ -intercepts by setting $f(x) = 0$ :
$-4(x+4)(x-3)^3 = 0$
The solutions are:
$x+4 = 0 \quad \Rightarrow \quad x = -4$ $(x-3)^3 = 0 \quad \Rightarrow \quad x = 3$
Thus, the $x$ -intercepts are $x = -4$ and $x = 3$ .

STEP 6

Find the $y$ -intercept by evaluating $f(0)$ :
$f(0) = -4(0+4)(0-3)^3$ $f(0) = -4(4)(-27)$ $f(0) = 432$
The $y$ -intercept is $y = 432$ .

STEP 7

Determine the real zeros and their multiplicities:
- $x = -4$ is a zero of multiplicity 1 (since it appears once in the factor $(x+4)$ ). - $x = 3$ is a zero of multiplicity 3 (since it appears three times in the factor $(x-3)^3$ ).

STEP 8

Based on multiplicity:
- The graph crosses the $x$ -axis at $x = -4$ because the multiplicity is odd. - The graph touches the $x$ -axis at $x = 3$ because the multiplicity is odd, but since it's a cubic term, it will also cross.

STEP 9

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Studdy Solution

STEP 1

1. The polynomial function is given by f(x)=−4(x+4)(x−3)3 f(x) = -4(x+4)(x-3)^3 f(x)=−4(x+4)(x−3)3.2. We need to analyze the polynomial based on its end behavior, intercepts, zeros, and graph.

STEP 2

1. Determine the end behavior of the polynomial.2. Find the x x x-intercepts and y y y-intercepts.3. Determine the real zeros and their multiplicities.4. Use a graphing utility to visualize the function.

STEP 3

STEP 4

STEP 5

STEP 6

Find the y y y-intercept by evaluating f(0) f(0) f(0):f(0)=−4(0+4)(0−3)3 f(0) = -4(0+4)(0-3)^3 f(0)=−4(0+4)(0−3)3 f(0)=−4(4)(−27) f(0) = -4(4)(-27) f(0)=−4(4)(−27) f(0)=432 f(0) = 432 f(0)=432The y y y-intercept is y=432 y = 432 y=432.

STEP 7

Determine the real zeros and their multiplicities:- x=−4 x = -4 x=−4 is a zero of multiplicity 1 (since it appears once in the factor (x+4) (x+4) (x+4)). - x=3 x = 3 x=3 is a zero of multiplicity 3 (since it appears three times in the factor (x−3)3 (x-3)^3 (x−3)3).

STEP 8

Based on multiplicity:- The graph crosses the x x x-axis at x=−4 x = -4 x=−4 because the multiplicity is odd. - The graph touches the x x x-axis at x=3 x = 3 x=3 because the multiplicity is odd, but since it's a cubic term, it will also cross.

STEP 9

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1. The polynomial function is given by $f(x) = -4(x+4)(x-3)^3$ .
2. We need to analyze the polynomial based on its end behavior, intercepts, zeros, and graph.

1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts and $y$ -intercepts.
3. Determine the real zeros and their multiplicities.
4. Use a graphing utility to visualize the function.

Find the $y$ -intercept by evaluating $f(0)$ :
$f(0) = -4(0+4)(0-3)^3$ $f(0) = -4(4)(-27)$ $f(0) = 432$
The $y$ -intercept is $y = 432$ .

Determine the real zeros and their multiplicities:
- $x = -4$ is a zero of multiplicity 1 (since it appears once in the factor $(x+4)$ ). - $x = 3$ is a zero of multiplicity 3 (since it appears three times in the factor $(x-3)^3$ ).

Based on multiplicity:
- The graph crosses the $x$ -axis at $x = -4$ because the multiplicity is odd. - The graph touches the $x$ -axis at $x = 3$ because the multiplicity is odd, but since it's a cubic term, it will also cross.