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Chapter 3: Problem 12

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$f(x)=5^{x}$$

Short Answer

Expert verified
The graph of the function \(f(x) = 5^x\) is an increasing curve, touching but not crossing the \(x\)-axis at negative infinity and rising rapidly as \(x\) increases. This can be verified by plotting the points (-2, 0.04), (-1, 0.2), (0, 1), (1, 5), and (2, 25) and by using a graphing utility.

Step by step solution

01

Understand the function

Firstly, the given function is \(f(x)=5^{x}\). This is an exponential function where 5 is the base and \(x\) is the power. When the base is greater than 1, the function increases as \(x\) increases, and it tends towards 0 as \(x\) goes towards negative infinity.
02

Create a table for coordinates

Create a table of values by choosing a range of \(x\) values and calculating the corresponding \(f(x)\) values. As an example, you might choose \(x = -2, -1, 0, 1, 2\). Calculate \(f(x)\) values by substituting the \(x\) values in the function, which results in \(f(-2) = 0.04, f(-1) = 0.2, f(0) = 1, f(1) = 5, f(2) = 25\).
03

Plot the function

Plot these points on the graph. Draw a smooth curve through these points. Remember that this graph should approach the \(x\)-axis as \(x\) goes towards negative infinity and rise rapidly as \(x\) increases.
04

Verify using a graphing utility

Use a reliable graphing utility to plot the function and confirm if the sketched graph is accurate. The plotted graph should match with the hand-drawn graph.

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Most popular questions from this chapter

Exercises \(153-155\) will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 a. Find Hungary's population, in millions, for 2006,2007 \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$3^{x+1}=9$$

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the natural base \(e\) when determining how much money I'd have in a bank account that earns compound interest subject to continuous compounding.

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts \((a)-(c) .\) Try generalizing this observation.
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