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Chapter 3: Problem 30

Use a graphing utility to graph the exponential function. $$y=3^{-|x|}$$

Short Answer

Expert verified
The graph of the function \(y=3^{-|x|}\) forms a peak at x = 0 with y = 1 and decays outwards to y = 0 as x approaches positive or negative infinity.

Step by step solution

01

Understanding the Exponential Function

The function \(y=3^{-|x|}\) can be interpreted as: For any x-value, take the absolute value of x, negate it and use it as an exponent of 3. Because of the absolute value operation, the function is symmetric with respect to the y-axis.
02

Plotting Points Manually

Though the exercise requires using a graphing utility, it may be beneficial to plug in a few x-values to understand what the graph might look like. Since the function is symmetric with respect to the y-axis, plotting values for -2, -1, 0, 1, 2 will suffice. The corresponding y-values are \(y=3^{-|-2|}=3^{-2}=0.1111\), \(y=3^{-|-1|}=3^{-1}=0.3333\), \(y=3^{-|0|}=3^{-0}=1\), \(y=3^{-|1|}=3^{-1}=0.3333\), and \(y=3^{-|2|}=3^{-2}=0.1111\). Plotting these points gives an idea of how the function behaves.
03

Using a Graphing Utility

Now that there's an understanding of how the function behaves, the next step is to input it into a graphing utility. Simply type in 'y=3^(-abs(x))' in the graphing utility and the graph of the function will be plotted.
04

Observing the Graph

The graph reveals a central peak with y-value of 1 at x = 0. As x approaches infinity or negative infinity, y decreases to 0. This reveals the rate at which the function converges to the x-axis.

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Most popular questions from this chapter

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=5.8\).

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2} / 450}, 70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log (3 x+4)=\log (x-10)$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 3 \\ \hline y & 1 & \frac{1}{4} \\ \hline \end{array} $$
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