Questions: Begin by graphing f(x)=2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x)=2^x-2 Which transformation is needed to graph the function g(x)=2^x-2 ? Choose the correct answer below. A. The graph of f(x)=2^x should be shifted 2 units downward. B. The graph of f(x)=2^x should be shifted 2 units upward. C. The graph of f(x)=2^x should be shifted 2 units to the right. D. The graph of f(x)=2^x should be shifted 2 units to the left. Graph g(x)=2^x-2 and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line. The equation of the asymptote for g(x)=2^x-2 is . (Type an equation.) The domain of g(x)=2^x-2 is . (Type your answer in interval notation.) The range of g(x)=2^x-2 is .

Begin by graphing f(x)=2^x. Then use transformations of this graph to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x)=2^x-2 Which transformation is needed to graph the function g(x)=2^x-2 ? Choose the correct answer below. A. The graph of f(x)=2^x should be shifted 2 units downward. B. The graph of f(x)=2^x should be shifted 2 units upward. C. The graph of f(x)=2^x should be shifted 2 units to the right. D. The graph of f(x)=2^x should be shifted 2 units to the left. Graph g(x)=2^x-2 and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line. The equation of the asymptote for g(x)=2^x-2 is . (Type an equation.) The domain of g(x)=2^x-2 is . (Type your answer in interval notation.) The range of g(x)=2^x-2 is .
Transcript text: Begin by graphing $f(x)=2^{x}$. Then use transformations of this graph to graph the given function. Be sure to graph and give the equation of the asymptote. Use the graph to determine the function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. \[ g(x)=2^{x-2} \] Which transformation is needed to graph the function $\mathrm{g}(\mathrm{x})=2^{\mathrm{x}-2}$ ? Choose the correct answer below. A. The graph of $f(x)=2^{x}$ should be shifted 2 units downward. B. The graph of $f(x)=2^{x}$ should be shifted 2 units upward. C. The graph of $f(x)=2^{x}$ should be shifted 2 units to the right. D. The graph of $f(x)=2^{x}$ should be shifted 2 units to the left. Graph $\mathrm{g}(\mathrm{x})=2^{\mathrm{x}-2}$ and its asymptote. Use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line. The equation of the asymptote for $g(x)=2^{x-2}$ is $\square$ . (Type an equation.) The domain of $g(x)=2^{x-2}$ is $\square$ 7. (Type your answer in interval notation.) The range of $g(x)=2^{x-2}$ is $\square$ $\square$.
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Solution

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Solution Steps

Step 1: Identifying the transformation

The function g(x) = 2x - 2 is a transformation of the function f(x) = 2x. The difference between the two functions is the subtraction of 2 from g(x). This corresponds to a vertical shift downwards.

Step 2: Determining the direction and magnitude of the shift

Since we are subtracting 2, the shift is 2 units downwards.

Step 3: The equation of the asymptote

The horizontal asymptote of f(x) = 2x is y = 0. Shifting the graph down by 2 units also shifts the asymptote down by 2 units. The new asymptote for g(x) = 2x - 2 is y = -2.

Final Answer:

The graph of f(x) = 2x should be shifted 2 units downward. The equation of the asymptote is y = -2.

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