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Magazine Chapter 4: Problem 9
Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$g(x)=3 e^{x}$$
Short Answer
Expert verified
Create a table of values and plot to display exponential growth.
Step by step solution
01
Choose input values (x values)
Select a range of \(x\) values to calculate \(g(x) = 3e^x\). You could start with small integer values such as \(-2, -1, 0, 1, 2\) to see the general shape of the graph.
02
Calculate the output values (g(x) values)
Using the function \(g(x) = 3e^x\), calculate the corresponding \(g(x)\) values for each chosen \(x\). For example, find \(g(-2)\), \(g(-1)\), \(g(0)\), \(g(1)\), and \(g(2)\).
03
Compute \(g(-2)\)
Calculate \[ g(-2) = 3e^{-2} = 3 imes \frac{1}{e^2} \approx 3 imes 0.1353 \approx 0.4053 \]
04
Compute \(g(-1)\)
Calculate \[ g(-1) = 3e^{-1} = 3 imes \frac{1}{e} \approx 3 imes 0.3679 \approx 1.1037 \]
05
Compute \(g(0)\)
Calculate \[ g(0) = 3e^{0} = 3 imes 1 = 3 \]
06
Compute \(g(1)\)
Calculate \[ g(1) = 3e^{1} = 3 imes e \approx 3 imes 2.7183 \approx 8.1549 \]
07
Compute \(g(2)\)
Calculate \[ g(2) = 3e^{2} = 3 imes e^2 \approx 3 imes 7.3891 \approx 22.1673 \]
08
Make a table of values
Organize the calculated values into a table:\[\begin{array}{c|c} x & g(x) \\hline-2 & 0.4053 \-1 & 1.1037 \0 & 3 \1 & 8.1549 \2 & 22.1673 \\end{array}\]
09
Sketch the graph
Plot the points from the table on a coordinate plane. Join these points with a smooth curve. The graph will show an exponential growth where the y-values increase rapidly as x increases.
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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 is an important concept in mathematics and helps us to visually analyze how values change rapidly. In the function \(g(x) = 3e^x\), the graph depicts how the function grows as \(x\) increases.
To start graphing, consider two important aspects of exponential functions:
By identifying these parts, we know the graph will increase exponentially. Start by plotting calculated points from prior calculations, then smoothly connect these points to form an exponential curve. Notice that as \(x\) becomes negative, the graph approaches but never touches the x-axis, demonstrating the asymptotic behavior of the function.
To start graphing, consider two important aspects of exponential functions:
- The base of the exponential, which is \(e\), a mathematical constant approximately equal to 2.71828.
- The coefficient before the exponential term, which is 3 in this scenario, indicating a vertical stretch of the graph.
By identifying these parts, we know the graph will increase exponentially. Start by plotting calculated points from prior calculations, then smoothly connect these points to form an exponential curve. Notice that as \(x\) becomes negative, the graph approaches but never touches the x-axis, demonstrating the asymptotic behavior of the function.
Using a Table of Values
Using a table of values is a powerful method to gain insights into the behavior of a function. By inputting selected \(x\) values into the function \(g(x) = 3e^x\), you can determine corresponding \(y\)-values.
Here's how to create and use a table:
This table assists in accurately plotting the function on a graph and understanding increments. This method ensures that each stage of the function is understood logically before visualizing and aids in identifying any mistakes in calculations possibly made when graphing.
Here's how to create and use a table:
- Select a range of input values \(x\), such as \(-2, -1, 0, 1, 2\), to encompass both negative and positive values.
- Calculate \(g(x)\) for each \(x\). This reveals different points on the graph, exposing how the function behaves at each point.
- Organize these values into pairs \((x, g(x))\), making a tidy table.
This table assists in accurately plotting the function on a graph and understanding increments. This method ensures that each stage of the function is understood logically before visualizing and aids in identifying any mistakes in calculations possibly made when graphing.
Exponential Growth
Exponential growth describes how quantities increase over time in a multiplicative process. The function \(g(x) = 3e^x\) is a classic example of exponential growth. Exponential growth is unique as it becomes more rapid as the independent variable \(x\) increases.
In mathematical terms, because \(e\) (Euler's number) is greater than one, \(e^x\) increases rapidly, making any function containing it grow exponentially.
Visualizing exponential growth helps in comprehending how minor shifts might result in significant differences down the line, making this a crucial concept in both math and various applications outside academics.
In mathematical terms, because \(e\) (Euler's number) is greater than one, \(e^x\) increases rapidly, making any function containing it grow exponentially.
- For any positive \(x\), \(g(x)\) rises continually, depicting acceleration in growth rate.
- If \(x\) is negative, the function still allows for small values but increases towards zero quickly.
- Practical applications include population growth, compound interest, and other real-world instances where small changes multiply over time.
Visualizing exponential growth helps in comprehending how minor shifts might result in significant differences down the line, making this a crucial concept in both math and various applications outside academics.
