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Chapter 12: Problem 4

Graph each exponential function. $$ y=3^{x}-1 $$

Short Answer

Expert verified
Plot points for selected \(x\) values and connect them smoothly.

Step by step solution

01

Understand the Exponential Function

In the given function, \(y = 3^x - 1\), the base is \(3\), and the exponent is \(x\). This is an exponential function, which means it will grow rapidly as \(x\) increases. The \(-1\) indicates a vertical shift downwards by 1 unit.
02

Create a Table of Values

To graph the function, we start by choosing a few values for \(x\) to compute the corresponding \(y\) values. For example, choose \(x = -1, 0, 1, 2\).- If \(x = -1\), \(y = 3^{-1} - 1 = \frac{1}{3} - 1 = -\frac{2}{3}\).- If \(x = 0\), \(y = 3^0 - 1 = 1 - 1 = 0\).- If \(x = 1\), \(y = 3^1 - 1 = 3 - 1 = 2\).- If \(x = 2\), \(y = 3^2 - 1 = 9 - 1 = 8\).These points are \((-1, -\frac{2}{3})\), \((0, 0)\), \((1, 2)\), and \((2, 8)\).
03

Plot the Points on the Graph

Using the table of values, plot the points on a coordinate plane: \((-1, -\frac{2}{3})\), \((0, 0)\), \((1, 2)\), and \((2, 8)\). These points help to form the shape of the graph.
04

Draw the Exponential Curve

Connect the plotted points with a smooth curve to demonstrate the exponential growth pattern. Make sure the graph approaches the horizontal asymptote at \(y = -1\) as \(x\) goes to negative infinity.
05

Analyze the Graph

The graph of \(y = 3^x - 1\) will pass through the points mentioned and will show exponential growth, with the line never crossing below the horizontal asymptote \(y = -1\). The domain is all real numbers, and the range is \(y > -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Exponential Functions
When graphing exponential functions, it's important to understand the basic shape and behavior. An exponential function follows the form \(y = a^x + b\), where \(a\) is the base of the exponent, \(x\) is the exponent, and \(b\) represents any vertical shifts in the graph. In our example \(y = 3^x - 1\), the base is \(3\) which means the function will grow quickly as \(x\) increases.
To correctly graph this function, you need to:
  • Identify the base, which tells you the direction and steepness of the curve.
  • Consider any shifts, like the \

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Most popular questions from this chapter

\(\log _{3} 10\) is between which two integers? Explain your answer.

Solve. $$ \log _{x} 49=2 $$

Solve each equation. $$ \log _{3} x+\log _{3}(x-8)=2 $$

The formula \(y=y_{0} e^{k t}\) gives the population size y of a population that experiences a relative growth rate \(k(k\) is positive if growth is increasing and \(k\) is negative if growth is decreasing). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time \(0 .\) Use this formula to solve Exercises 55 and \(56 .\) Round answers to the nearest year. (Source for data: U.S. Census Bureau and Federal Reserve Bank of Chicago) In \(2009,\) the population of Michigan was approximately 9,970,000 and decreasing according to the formula \(y=y_{0} e^{-0.003 t}\). Assume that the population continues to decrease according to the given formula and predict how many years after which the population of Michigan will be \(9,500,000 .\) (Hint: Let \(y_{0}=9,970,000 ; y=9,500,000\), and solve for \(t\).)

Solve. $$ 3^{\log _{3} 5}=x $$
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