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Magazine Chapter 4: Problem 13
Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$g(x)=3(1.3)^{x}$$
Short Answer
Expert verified
Plot points from x = -2 to 3 using g(x)=3(1.3)^x, draw a smooth increasing curve.
Step by step solution
01
Understanding the Function
The function given is an exponential function of the form \(g(x)=3(1.3)^{x}\). This means the base of our function is 1.3, and the leading coefficient is 3.
02
Choose Values for x
Select a set of x-values to use in creating our table. A good range might be from -2 to 3 to see how the function behaves for negative and positive x-values.
03
Calculate Corresponding g(x) Values
Use the function formula to calculate g(x) for each chosen value of x. For example, for \(x = -2\), it is \(g(-2) = 3(1.3)^{-2}\), for \(x = 0\), it is \(g(0) = 3(1.3)^{0}\), and so on.
04
Fill Out the Table of Values
Create a table by calculating the y-values calculated using the exponential function. For example:
- x = -2, g(x) 鈮� 1.78
- x = -1, g(x) 鈮� 2.31
- x = 0, g(x) = 3
- x = 1, g(x) 鈮� 3.9
- x = 2, g(x) 鈮� 5.07
- x = 3, g(x) 鈮� 6.59
05
Plot the Points on Graph
Using the table, plot the points \((-2, 1.78), (-1, 2.31), (0, 3), (1, 3.9), (2, 5.07), (3, 6.59)\) on graph paper or graphing software.
06
Draw the Curve
Connect the plotted points with a smooth curve to represent the graph of the function. Ensure the curve reflects the exponential growth characteristic, starting from below the y-axis (for negative x) and rising steeply for positive x.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Graphing Techniques for Exponential Functions
Graphing exponential functions, like \(g(x)=3(1.3)^{x}\), requires a clear understanding of how these functions behave. Exponential functions are characterized by a constant base raised to a variable exponent. This structure causes them to have unique graph traits:
- The function always crosses the y-axis at the point equal to the leading coefficient. Here, when \(x=0\), \(g(0)=3\).
- These graphs typically exhibit exponential growth or decay. The base value greater than 1 (1.3, in this case) implies growth.
- The shape has a distinct curve: it increases slowly for small x-values and speeds up with larger x-values.
- The graph never touches the x-axis but approaches it as x decreases.
- Begin by plotting points from a table of values calculated from your function.
- Use both negative and positive values of x to capture the curve's behavior fully.
- Smoothly connect these points to outline the path of the exponential function.
Using a Function Table to Determine Plot Points
Creating a function table is a practical method for sketching the graph of an exponential function. In the case of \(g(x)=3(1.3)^{x}\), a table of values acts as a guide for plotting. Here's how it works:
- Choose a range of x-values. The range \(-2\) to \(3\) was selected to show both ends of the growth spectrum.
- Calculate the corresponding y-values, or \(g(x)\), by substituting each x-value into the function.
- For instance:
- When \(x=-2\), \(g(-2)=3(1.3)^{-2}\approx 1.78\).
- For \(x=0\), \(g(0)=3\), since any number raised to the power of zero equals 1.
- Continue this for each x-value to complete the table, such as \(x=1\) giving \(g(1)\approx 3.9\), and so forth.
Exploring the Concept of Exponential Growth
Exponential growth is a key feature of the function \(g(x)=3(1.3)^{x}\). It can be understood through its equation and graphical representation:
- Exponential growth occurs when the base of the exponential function is greater than 1. In our function, \(1.3\) facilitates this, leading to continuous growth as x increases.
- Unlike linear growth, which adds a constant amount, exponential growth multiplies the current amount. Hence, the rapid rise extends with larger x-values.
- The rate of increase reveals itself in the steepness of the graph. Initially, for small x-values, increases might look minor, but the rate compounds quickly, reflecting real-world phenomena such as population growth.
- Notice how, from the function table, even a modest jump from \(x=1\) to \(x=2\) sees a notable rise from \(3.9\) to \(5.07\).
- This amplifying effect continues with higher values of x, a typical characteristic of exponential growth.
