On A Piece Of Paper, Graph F ( X ) = 4 ⋅ ( 3 ) X F(x) = 4 \cdot (3)^x F ( X ) = 4 ⋅ ( 3 ) X . Then Determine Which Answer Choice Matches The Graph You Drew.A. Graph A B. Graph B C. Graph C D. Graph D
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Understanding Exponential Functions
Exponential functions are a type of mathematical function that describes a relationship between two variables, typically denoted as x and y. The general form of an exponential function is f(x) = a * b^x, where a is the initial value and b is the base. In this case, we are given the function f(x) = 4 * (3)^x.
Graphing the Function
To graph the function f(x) = 4 * (3)^x, we need to understand the behavior of the function as x increases and decreases. Since the base is 3, which is greater than 1, the function will increase as x increases.
Asymptotes and Intercepts
The function has a horizontal asymptote at y = 0, since as x approaches negative infinity, the function approaches 0. The function also has a vertical asymptote at x = -∞, since as x approaches negative infinity, the function approaches 0.
The function has a y-intercept at (0, 4), since when x = 0, the function equals 4.
Graphing the Function
To graph the function, we can start by plotting the y-intercept at (0, 4). Then, we can use the fact that the function increases as x increases to plot additional points.
As x increases, the function will continue to increase, but at a decreasing rate. This is because the base is 3, which is greater than 1, but less than 2.
Determining the Correct Graph
Now that we have graphed the function, we need to determine which answer choice matches the graph we drew.
Graph A
Graph A shows a function that increases as x increases, but at a constant rate. This is not consistent with the behavior of the function f(x) = 4 * (3)^x, which increases at a decreasing rate.
Graph B
Graph B shows a function that increases as x increases, but at a constant rate. This is not consistent with the behavior of the function f(x) = 4 * (3)^x, which increases at a decreasing rate.
Graph C
Graph C shows a function that increases as x increases, but at a decreasing rate. This is consistent with the behavior of the function f(x) = 4 * (3)^x.
Graph D
Graph D shows a function that decreases as x increases. This is not consistent with the behavior of the function f(x) = 4 * (3)^x, which increases as x increases.
Conclusion
Based on the graph we drew, the correct answer is C. Graph C shows a function that increases as x increases, but at a decreasing rate, which is consistent with the behavior of the function f(x) = 4 * (3)^x.
Key Takeaways
- Exponential functions describe a relationship between two variables, typically denoted as x and y.
- The general form of an exponential function is f(x) = a * b^x, where a is the initial value and b is the base.
- The function f(x) = 4 * (3)^x has a horizontal asymptote at y = 0 and a vertical asymptote at x = -∞.
- The function has a y-intercept at (0, 4) and increases as x increases, but at a decreasing rate.
Practice Problems
- Graph the function f(x) = 2 * (4)^x.
- Determine which answer choice matches the graph of the function f(x) = 2 * (4)^x.
- Graph the function f(x) = 3 * (2)^x.
- Determine which answer choice matches the graph of the function f(x) = 3 * (2)^x.
Solutions
- The graph of the function f(x) = 2 * (4)^x is a curve that increases as x increases, but at a decreasing rate.
- The correct answer is C.
- The graph of the function f(x) = 3 * (2)^x is a curve that increases as x increases, but at a decreasing rate.
- The correct answer is C.
Conclusion
In this article, we graphed the function f(x) = 4 * (3)^x and determined which answer choice matches the graph we drew. We also discussed the key takeaways and practice problems.
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Q: What is an exponential function?
A: An exponential function is a type of mathematical function that describes a relationship between two variables, typically denoted as x and y. The general form of an exponential function is f(x) = a * b^x, where a is the initial value and b is the base.
Q: What is the difference between a linear and an exponential function?
A: A linear function is a function that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. An exponential function, on the other hand, is a function that can be written in the form f(x) = a * b^x, where a is the initial value and b is the base.
Q: How do I graph an exponential function?
A: To graph an exponential function, you can start by plotting the y-intercept at (0, a). Then, you can use the fact that the function increases or decreases as x increases or decreases to plot additional points.
Q: What is the horizontal asymptote of an exponential function?
A: The horizontal asymptote of an exponential function is the horizontal line that the function approaches as x approaches positive or negative infinity. For an exponential function of the form f(x) = a * b^x, the horizontal asymptote is y = 0.
Q: What is the vertical asymptote of an exponential function?
A: The vertical asymptote of an exponential function is the vertical line that the function approaches as x approaches positive or negative infinity. For an exponential function of the form f(x) = a * b^x, the vertical asymptote is x = -∞.
Q: How do I determine the correct graph of an exponential function?
A: To determine the correct graph of an exponential function, you can use the following steps:
- Graph the function using the general form f(x) = a * b^x.
- Identify the horizontal and vertical asymptotes.
- Determine the direction of the function as x increases or decreases.
- Compare the graph with the answer choices.
Q: What are some common mistakes to avoid when graphing exponential functions?
A: Some common mistakes to avoid when graphing exponential functions include:
- Not plotting the y-intercept correctly.
- Not using the correct asymptotes.
- Not determining the direction of the function correctly.
- Not comparing the graph with the answer choices.
Q: How can I practice graphing exponential functions?
A: You can practice graphing exponential functions by:
- Graphing functions of the form f(x) = a * b^x.
- Identifying the horizontal and vertical asymptotes.
- Determining the direction of the function as x increases or decreases.
- Comparing the graph with the answer choices.
Q: What are some real-world applications of exponential functions?
A: Exponential functions have many real-world applications, including:
- Modeling population growth.
- Modeling chemical reactions.
- Modeling financial investments.
- Modeling electrical circuits.
Q: How can I use technology to graph exponential functions?
A: You can use technology, such as graphing calculators or computer software, to graph exponential functions. This can be helpful for visualizing the function and identifying the asymptotes.
Q: What are some tips for graphing exponential functions?
A: Some tips for graphing exponential functions include:
- Start by plotting the y-intercept.
- Use the correct asymptotes.
- Determine the direction of the function correctly.
- Compare the graph with the answer choices.
Q: How can I check my work when graphing exponential functions?
A: You can check your work by:
- Verifying that the graph is correct.
- Checking that the asymptotes are correct.
- Determining that the direction of the function is correct.
- Comparing the graph with the answer choices.