Questions: Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in the figures below and, if necessary, the transformations of Section 1.3. y = 4^x - 2

Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in the figures below and, if necessary, the transformations of Section 1.3.

y = 4^x - 2

Transcript text: Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in the figures below and, if necessary, the transformations of Section 1.3. \[ y=4^{x}-2 \]

Solution

Solution Steps

Step 1: Identify the base function

The base function is \(y = 4^x\). This is an exponential function with a base greater than 1, so it increases as \(x\) increases. It also has a horizontal asymptote at \(y = 0\).

Step 2: Apply the vertical shift

The given function \(y = 4^x - 2\) is a vertical shift of the base function \(y = 4^x\) downwards by 2 units. This means every point on the graph of \(y = 4^x\) is moved down 2 units. The horizontal asymptote will also shift down by 2 units, becoming \(y = -2\).

Step 3: Sketch the graph

The graph will pass through the point \((0, 4^0 - 2) = (0, 1 - 2) = (0, -1)\). It will approach the horizontal asymptote \(y = -2\) as \(x\) approaches negative infinity, and it will increase rapidly as \(x\) approaches positive infinity.

Final Answer

The sketch of the graph should resemble an exponential growth curve shifted downwards. It has a horizontal asymptote at \(y=-2\) and passes through the point \((0, -1)\). It rises steeply to the right and approaches the asymptote to the left. I cannot draw here, but a rough sketch would depict this behavior. You can use the graph of \(y = 4^x\) provided, and shift it down two units. \(\boxed{\text{See the description above.}}\)

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