Solved on Jan 19, 2024

Analyze the end behavior of $f(x) = 12x + 144 - 6x^2$ as $x \rightarrow \pm \infty$ .

STEP 1

Assumptions
1. The function given is $f(x) = 12x + 144 - 6x^2$ .
2. The end behavior of a function describes what happens to the function values as $x$ approaches positive or negative infinity.
3. The leading term of the polynomial function determines its end behavior.

STEP 2

Identify the leading term of the polynomial function.
The leading term is the term with the highest power of $x$ . For the function $f(x) = 12x + 144 - 6x^2$ , the leading term is $-6x^2$ .

STEP 3

Determine the degree and the coefficient of the leading term.
The leading term is $-6x^2$ , so the degree is $2$ and the coefficient is $-6$ .

STEP 4

Analyze the end behavior based on the leading term.
For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise to infinity. If the leading coefficient is negative, both ends of the graph fall to negative infinity.

STEP 5

Apply the end behavior analysis to the given function.
Since the leading coefficient of $f(x)$ is $-6$ (which is negative) and the degree is $2$ (which is even), both ends of the graph of $f(x)$ will fall to negative infinity.

STEP 6

Write the end behavior of the function $f(x)$ .
As $x \rightarrow -\infty$ , $f(x) \rightarrow -\infty$ and as $x \rightarrow \infty$ , $f(x) \rightarrow -\infty$ .
The correct answer is: as $x \rightarrow -\infty, f(x) \rightarrow -\infty$ and as $x \rightarrow \infty, f(x) \rightarrow -\infty$ .

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Analyze the end behavior of $f(x) = 12x + 144 - 6x^2$ as $x \rightarrow \pm \infty$ .

STEP 1

Assumptions
1. The function given is $f(x) = 12x + 144 - 6x^2$ .
2. The end behavior of a function describes what happens to the function values as $x$ approaches positive or negative infinity.
3. The leading term of the polynomial function determines its end behavior.

STEP 2

Identify the leading term of the polynomial function.
The leading term is the term with the highest power of $x$ . For the function $f(x) = 12x + 144 - 6x^2$ , the leading term is $-6x^2$ .

STEP 3

Determine the degree and the coefficient of the leading term.
The leading term is $-6x^2$ , so the degree is $2$ and the coefficient is $-6$ .

STEP 4

Analyze the end behavior based on the leading term.
For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise to infinity. If the leading coefficient is negative, both ends of the graph fall to negative infinity.

STEP 5

Apply the end behavior analysis to the given function.
Since the leading coefficient of $f(x)$ is $-6$ (which is negative) and the degree is $2$ (which is even), both ends of the graph of $f(x)$ will fall to negative infinity.

STEP 6

Write the end behavior of the function $f(x)$ .
As $x \rightarrow -\infty$ , $f(x) \rightarrow -\infty$ and as $x \rightarrow \infty$ , $f(x) \rightarrow -\infty$ .
The correct answer is: as $x \rightarrow -\infty, f(x) \rightarrow -\infty$ and as $x \rightarrow \infty, f(x) \rightarrow -\infty$ .

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Analyze the end behavior of f(x)=12x+144−6x2f(x) = 12x + 144 - 6x^2f(x)=12x+144−6x2 as x→±∞x \rightarrow \pm \inftyx→±∞.

STEP 1

Assumptions1. The function given is f(x)=12x+144−6x2f(x) = 12x + 144 - 6x^2f(x)=12x+144−6x2.2. The end behavior of a function describes what happens to the function values as xxx approaches positive or negative infinity.3. The leading term of the polynomial function determines its end behavior.

STEP 2

Identify the leading term of the polynomial function.The leading term is the term with the highest power of xxx. For the function f(x)=12x+144−6x2f(x) = 12x + 144 - 6x^2f(x)=12x+144−6x2, the leading term is −6x2-6x^2−6x2.

STEP 3

Determine the degree and the coefficient of the leading term.The leading term is −6x2-6x^2−6x2, so the degree is 222 and the coefficient is −6-6−6.

STEP 4

Analyze the end behavior based on the leading term.For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise to infinity. If the leading coefficient is negative, both ends of the graph fall to negative infinity.

STEP 5

Apply the end behavior analysis to the given function.Since the leading coefficient of f(x)f(x)f(x) is −6-6−6 (which is negative) and the degree is 222 (which is even), both ends of the graph of f(x)f(x)f(x) will fall to negative infinity.

STEP 6

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Analyze the end behavior of $f(x) = 12x + 144 - 6x^2$ as $x \rightarrow \pm \infty$ .

Assumptions
1. The function given is $f(x) = 12x + 144 - 6x^2$ .
2. The end behavior of a function describes what happens to the function values as $x$ approaches positive or negative infinity.
3. The leading term of the polynomial function determines its end behavior.

Identify the leading term of the polynomial function.
The leading term is the term with the highest power of $x$ . For the function $f(x) = 12x + 144 - 6x^2$ , the leading term is $-6x^2$ .

Determine the degree and the coefficient of the leading term.
The leading term is $-6x^2$ , so the degree is $2$ and the coefficient is $-6$ .

Analyze the end behavior based on the leading term.
For even-degree polynomials, if the leading coefficient is positive, both ends of the graph rise to infinity. If the leading coefficient is negative, both ends of the graph fall to negative infinity.

Apply the end behavior analysis to the given function.
Since the leading coefficient of $f(x)$ is $-6$ (which is negative) and the degree is $2$ (which is even), both ends of the graph of $f(x)$ will fall to negative infinity.