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Chapter 13: Problem 12

Graph each exponential function. $$y=3^{x+2}$$

Short Answer

Expert verified
To graph \(y = 3^{x+2}\), first analyze the function and identify the base and the constant. Calculate key points, such as (-2, 1), (-1, 3), (0, 9), (1, 27), and (2, 81). Determine the y-intercept, (0, 9), and the horizontal asymptote, \(y = 0\). Sketch the graph by plotting these points, drawing a smooth curve, and adding the horizontal asymptote. The graph showcases the exponential curve starting from the horizontal asymptote and then shooting up.

Step by step solution

01

Analyze the function

An exponential function is of the form \(y = ab^x\), where \(a\) is a constant, \(b > 0\), and \(b \neq 1\). In our given function, \(y = 3^{x+2}\), we can view it as \(y = 3^x \cdot 3^2\). So, the base \(b = 3\), and the constant \(a = 3^2 = 9\). The graph of this exponential function will have the same general shape as the graph of the basic exponential function \(y = 3^x\), but it will be transformed because of the constant factor of 9 and the shift in the exponent.
02

Calculate some key points

We will compute the function values for some important points. These points will help us understand the behavior at crucial intervals: - When \(x = -2\), the function becomes \(y = 3^0 = 1\) - When \(x = -1\), the function becomes \(y = 3^1 = 3\) - When \(x = 0\), the function becomes \(y = 3^2 = 9\) - When \(x = 1\), the function becomes \(y = 3^3 = 27\) - When \(x = 2\), the function becomes \(y = 3^4 = 81\)
03

Determine the y-intercept and horizontal asymptote

The y-intercept is the point where the graph crosses the y-axis, i.e., when \(x = 0\). From our calculations in Step 2, when \(x = 0\), \(y = 9\). So, the y-intercept is at the point (0, 9). The horizontal asymptote is a horizontal line that the graph approaches but never crosses as the input value becomes very small (negative). For positive base exponential functions, the horizontal asymptote is always at \(y = 0\), because the function approaches 0 as \(x\) becomes very large negatively.
04

Sketch the graph

Now that we have some key points and features, we can sketch the graph of the function \(y = 3^{x+2}\): 1. Begin by plotting the y-intercept at the point (0, 9). 2. Next, plot the points we calculated in Step 2: (-2, 1), (-1, 3), (1, 27), and (2, 81). 3. Draw a smooth curve that passes through all these points. 4. Add the horizontal asymptote at \(y = 0\). By following these steps, we should obtain a graph of the exponential function \(y = 3^{x+2}\), which showcases the exponential curve starting from the horizontal asymptote and then shooting up.

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Key Concepts

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

Graphing Exponential Functions
Graphing exponential functions involves visualizing the function on a coordinate plane. For a function like \(y = 3^{x+2}\), it helps to start by identifying its key components. An exponential function, in general, is in the form \(y = ab^x\), where \(b\) (the base) is a positive number not equal to 1, and \(a\) is a constant.
In our case, we've re-framed \(y = 3^{x+2}\) as \(y = 9 \times 3^x\) with \(a = 9\) and \(b = 3\). This means the graph retains the exponential shape but transforms due to the factor of 9 and an exponent shift.
To plot the graph:
  • Calculate and plot specific points for various \(x\) values.
  • Use these points to create a smooth curve.
  • Start at the y-intercept and ensure the curve approaches the horizontal asymptote.
The resulting graph will have a typical exponential rise as \(x\) increases.
Horizontal Asymptote
The concept of a horizontal asymptote is crucial in understanding exponential functions. For functions of the form \(y = b^x\), where \(b > 0\), the function approaches, but does not usually cross, a horizontal asymptote.
In the function \(y = 3^{x+2}\), the horizontal asymptote is at \(y = 0\). This indicates that as \(x\) becomes negatively large, \(3^{x+2}\) approaches zero.
This asymptotic behavior explains why the lower part of the graph settles close to the x-axis yet never touches it. Thus, seeing the horizontal asymptote helps you to understand how the graph behaves for extreme values of \(x\) and ensures accurate plotting of its shape.
Y-intercept
The y-intercept is the point where the graph intersects the y-axis, which means it's the point when \(x = 0\). To find the y-intercept of the function \(y = 3^{x+2}\), substitute 0 for \(x\). This results in: \[y = 3^{0+2} = 3^2 = 9\]Therefore, the y-intercept is at (0, 9).
This point is essential in graphing because it provides a reliable starting location to plot the graph of an exponential function. It helps in drawing the curve accurately as it emerges from the y-axis.
Transformation of Exponential Functions
Transformation in exponential functions involves shifting or scaling the graph from its basic form. The function \(y = 3^{x+2}\) can be broken down to help understand its transformation.
Firstly, the expression \(x + 2\) in the exponent denotes a horizontal shift. Specifically, each point on the function \(y = 3^x\) shifts to the left by 2 units.
Secondly, the multiplication by a constant, \(a = 9\), results in a vertical stretch. Instead of starting at 1 (as in \(y = 3^x\)), this exponential curve starts higher.
Understanding these transformations helps in accurately sketching the graph and allows interpretation of how different components impact the graph's appearance.

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