Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Understanding and Writing Function Rules from Graphs
#### Instructions:
Use the graph to write the function rule. How do you know you are correct?
#### Graph Description:
The graph plotted here shows a curve which passes through specific points marked in red. The x and y axes are labeled, and the points where the curve intersects the y-axis and x-axis can provide valuable information about the function.
1. **Reading Points:**
- The graph in this image shows a curve passing through the points (0, 24), (1, 12), and (2, 6).
2. **Identifying Function Type:**
- The curve suggests an exponential decay function, often represented by the function rule of the form \( y = a \cdot b^x \), where 'a' is a constant and 'b' is the base of the exponent.
3. **Determining the Function Rule:**
- Given the points provided:
- When \( x = 0 \), \( y = 24 \). This tells us that \( a = 24 \) because any number raised to the power of 0 is 1.
- When \( x = 1 \), \( y = 12 \). Substituting these values into \( y = a \cdot b^x \) gives us \( 12 = 24 \cdot b \), solving for \( b \), we get \( b = 0.5 \).
Therefore, the function rule can be written as:
\[ y = 24 \cdot (0.5)^x \]
4. **Validation:**
- To confirm the rule, substitute \( x = 2 \) into the function \( y = 24 \cdot (0.5)^x \):
- \( y = 24 \cdot (0.5)^2 \)
- \( y = 24 \cdot 0.25 \)
- \( y = 6 \), which matches the point (2, 6) on the graph, confirming our function rule.
Using this process of observing points and deriving a consistent function rule allows you to determine the specific function represented by a graph.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F716b07bd-8c61-4cea-9832-6f6fb0ec5c5a%2F7000b29d-bf2d-4f9d-878d-98665cf7ff8f%2Fu6behbn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding and Writing Function Rules from Graphs
#### Instructions:
Use the graph to write the function rule. How do you know you are correct?
#### Graph Description:
The graph plotted here shows a curve which passes through specific points marked in red. The x and y axes are labeled, and the points where the curve intersects the y-axis and x-axis can provide valuable information about the function.
1. **Reading Points:**
- The graph in this image shows a curve passing through the points (0, 24), (1, 12), and (2, 6).
2. **Identifying Function Type:**
- The curve suggests an exponential decay function, often represented by the function rule of the form \( y = a \cdot b^x \), where 'a' is a constant and 'b' is the base of the exponent.
3. **Determining the Function Rule:**
- Given the points provided:
- When \( x = 0 \), \( y = 24 \). This tells us that \( a = 24 \) because any number raised to the power of 0 is 1.
- When \( x = 1 \), \( y = 12 \). Substituting these values into \( y = a \cdot b^x \) gives us \( 12 = 24 \cdot b \), solving for \( b \), we get \( b = 0.5 \).
Therefore, the function rule can be written as:
\[ y = 24 \cdot (0.5)^x \]
4. **Validation:**
- To confirm the rule, substitute \( x = 2 \) into the function \( y = 24 \cdot (0.5)^x \):
- \( y = 24 \cdot (0.5)^2 \)
- \( y = 24 \cdot 0.25 \)
- \( y = 6 \), which matches the point (2, 6) on the graph, confirming our function rule.
Using this process of observing points and deriving a consistent function rule allows you to determine the specific function represented by a graph.
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