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Chapter 3: Problem 34

Use a graphing utility to graph the exponential function. $$y=4^{x+1}-2$$

Short Answer

Expert verified
The graph of the function \(y = 4^{x+1} - 2\) can be obtained by first graphing the base function \(4^x\), then shifting it to the left by 1 unit and down by 2 units.

Step by step solution

01

Understand the Exponential Function

First, examine the exponential function given, which is \(y = 4^{x+1} - 2\). Here, \(4\) is the base, \((x + 1)\) is the power, and \(2\) is subtracted from the result of the exponential operation.
02

Identify the Graph Transformation

The base function is \(4^x\). Due to the presence of \(+1\) in the exponent, the graph will shift to the left by 1 unit. This is because \(+1\) is subtracted from \(x\), moving the graph left. The \(2\) is subtracted from the entire function, causing a shift down by 2 units.
03

Graph the Exponential Function

Next, utilize the graphing utility to graph the function. Begin with the base function \(4^x\) and shift it following the transformations identified in Step 2. Move the graph to the left by 1 unit and down by 2 units to get the graph of the given exponential function \(y = 4^{x+1} - 2\).

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln (x+5)=\ln (x-1)-\ln (x+1)$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$6 e^{1-x}=25$$

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window. \(f(x)=3 \ln x-1\)

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$8 e^{-2 x / 3}=11$$

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$
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