QuestionDescribe the end behavior of the graphs of the function. $f(x)=-4(2)^{x}-2$
As $x \rightarrow-\infty, f(x) \rightarrow$ $\square$ As $x \rightarrow \infty, f(x) \rightarrow$ $\square$

Studdy Solution

STEP 1

1. The function $f(x) = -4(2)^x - 2$ is an exponential function.
2. The base of the exponential function is $2$ , which is greater than $1$ .
3. The coefficient of the exponential term is negative, which affects the direction of the graph.
4. The end behavior of the function is determined by the behavior of the exponential term as $x$ approaches $-\infty$ and $\infty$ .

STEP 2

1. Analyze the behavior of the exponential term $(2)^x$ as $x \rightarrow -\infty$ .
2. Analyze the behavior of the exponential term $(2)^x$ as $x \rightarrow \infty$ .
3. Determine the end behavior of $f(x)$ as $x \rightarrow -\infty$ .
4. Determine the end behavior of $f(x)$ as $x \rightarrow \infty$ .

STEP 3

As $x \rightarrow -\infty$ , the term $(2)^x$ approaches $0$ . This is because any positive number raised to a large negative power approaches zero.

STEP 4

As $x \rightarrow \infty$ , the term $(2)^x$ grows without bound and approaches $\infty$ . This is because any positive number greater than $1$ raised to a large positive power increases indefinitely.

STEP 5

To determine the end behavior of $f(x)$ as $x \rightarrow -\infty$ :
$f(x) = -4(2)^x - 2$
Since $(2)^x \rightarrow 0$ , the term $-4(2)^x \rightarrow 0$ . Therefore,
$f(x) \rightarrow -2$
As $x \rightarrow -\infty, f(x) \rightarrow -2$ .

STEP 6

To determine the end behavior of $f(x)$ as $x \rightarrow \infty$ :
$f(x) = -4(2)^x - 2$
Since $(2)^x \rightarrow \infty$ , the term $-4(2)^x \rightarrow -\infty$ . Therefore,
$f(x) \rightarrow -\infty$
As $x \rightarrow \infty, f(x) \rightarrow -\infty$ .
The end behavior of the function is:
As $x \rightarrow -\infty, f(x) \rightarrow -2$ .
As $x \rightarrow \infty, f(x) \rightarrow -\infty$ .

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