Construct an exponential function to represent the graphed function below. y f(x) 8 4 -8 -6 -4 -2 4 8 -2 -4 -6 -8 6. 2. 2.
Construct an exponential function to represent the graphed function below. y f(x) 8 4 -8 -6 -4 -2 4 8 -2 -4 -6 -8 6. 2. 2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Construct an Exponential Function to Represent the Graphed Function Below
**Graph Explanation:**
The provided graph depicts an exponential function \( f(x) \).
**Axes:**
- The horizontal axis represents the \( x \)-axis, ranging from -8 to 8.
- The vertical axis represents the \( y \)-axis, ranging from -8 to 8.
**Graph Details:**
- The function \( f(x) \) is drawn in red.
- There are several black dots indicating specific points on the graph to denote key points, specifically at:
- (-2, 1)
- (-1, 1.5)
- (0, 2)
- (1, 2.7)
- (2, 3.4)
The curve shows an exponential growth pattern, where the function value increases slowly for negative and small positive values of \( x \), and then it rapidly rises as \( x \) becomes larger.
**Objective:**
Construct an exponential function that fits the given data points on the graph.
**Approach:**
An exponential function can be generally represented as:
\[ f(x) = a \cdot b^x \]
Where:
- \( a \) is the initial value (y-intercept),
- \( b \) is the base rate of growth (or decay).
From the graph, one can observe the y-intercept value when x = 0:
\[ f(0) = 2 \implies a = 2 \]
The value of \( b \) can be estimated by examining the growth pattern between points. For a more detailed and precise fitting, regression techniques or further calculation using logarithms might be required, but for an educational context, identifying the base incrementally can set a foundational understanding.
**Conclusion:**
Using the points provided, a basic exponential model can be hypothesized. Further adjustments based on more precise techniques can lead to an accurate function representing the observed data.
Keep in mind, displaying and understanding how to construct such functions is a crucial step in mastering exponential relationships in mathematical education.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F098f8e04-469a-4a6a-8094-b45e233f0633%2Fbc4685ec-308a-437d-b00d-04661be56e9d%2Fpn5513h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Construct an Exponential Function to Represent the Graphed Function Below
**Graph Explanation:**
The provided graph depicts an exponential function \( f(x) \).
**Axes:**
- The horizontal axis represents the \( x \)-axis, ranging from -8 to 8.
- The vertical axis represents the \( y \)-axis, ranging from -8 to 8.
**Graph Details:**
- The function \( f(x) \) is drawn in red.
- There are several black dots indicating specific points on the graph to denote key points, specifically at:
- (-2, 1)
- (-1, 1.5)
- (0, 2)
- (1, 2.7)
- (2, 3.4)
The curve shows an exponential growth pattern, where the function value increases slowly for negative and small positive values of \( x \), and then it rapidly rises as \( x \) becomes larger.
**Objective:**
Construct an exponential function that fits the given data points on the graph.
**Approach:**
An exponential function can be generally represented as:
\[ f(x) = a \cdot b^x \]
Where:
- \( a \) is the initial value (y-intercept),
- \( b \) is the base rate of growth (or decay).
From the graph, one can observe the y-intercept value when x = 0:
\[ f(0) = 2 \implies a = 2 \]
The value of \( b \) can be estimated by examining the growth pattern between points. For a more detailed and precise fitting, regression techniques or further calculation using logarithms might be required, but for an educational context, identifying the base incrementally can set a foundational understanding.
**Conclusion:**
Using the points provided, a basic exponential model can be hypothesized. Further adjustments based on more precise techniques can lead to an accurate function representing the observed data.
Keep in mind, displaying and understanding how to construct such functions is a crucial step in mastering exponential relationships in mathematical education.
**Graph Description:**
- The graph represents the function \( f(x) \) and is an exponential curve.
- The x-axis ranges from -8 to 8, and the y-axis ranges from -8 to 8.
- Key points plotted on the curve include: (-3, 0), (-2, 0), (-1, 0), (0, 1), (1, 2), (2, 5), (3, 9).
- The graph increases slowly for negative values of \( x \) and increases steeply for positive values of \( x \).
**Step-by-Step Construction:**
1. **Identify the Form of an Exponential Function:**
The general form of an exponential function is:
\[
f(x) = a \cdot b^x
\]
where \( a \) is a constant, \( b \) is the base of the exponential function.
2. **Determine the Base (b):**
To find the base \( b \), observe how the function value changes as \( x \) increases. We use some known points to identify this. Let's start with the base using the points where \( x = 0 \) and \( x = 1 \) since \( f(0) = 1 \) and \( f(1) = 2 \):
\[
f(0) = a \cdot b^0 = a \cdot 1 = a = 1
\]
\[
f(1) = 1 \cdot b^1 = b = 2
\]
Thus, \( b = 2 \).
3. **Construct the Function:**
Since we have found \( a = 1 \) and \( b = 2 \), the exponential function that represents the graph is:
\[
f(x) = 1 \cdot 2^x = 2^x
\]
**Explanation of Key Points:**
- The y-value increases very gradually for negative x-values (close to 0 but non-negative).
- For positive x-values, the y-value grows at an exponential rate, confirming the exponential behavior as described by \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F098f8e04-469a-4a6a-8094-b45e233f0633%2Fbc4685ec-308a-437d-b00d-04661be56e9d%2F403o8ph_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Construct an Exponential Function to Represent the Graphed Function
To construct an exponential function to represent the graphed function shown below, follow these steps:
**Graph Description:**
- The graph represents the function \( f(x) \) and is an exponential curve.
- The x-axis ranges from -8 to 8, and the y-axis ranges from -8 to 8.
- Key points plotted on the curve include: (-3, 0), (-2, 0), (-1, 0), (0, 1), (1, 2), (2, 5), (3, 9).
- The graph increases slowly for negative values of \( x \) and increases steeply for positive values of \( x \).
**Step-by-Step Construction:**
1. **Identify the Form of an Exponential Function:**
The general form of an exponential function is:
\[
f(x) = a \cdot b^x
\]
where \( a \) is a constant, \( b \) is the base of the exponential function.
2. **Determine the Base (b):**
To find the base \( b \), observe how the function value changes as \( x \) increases. We use some known points to identify this. Let's start with the base using the points where \( x = 0 \) and \( x = 1 \) since \( f(0) = 1 \) and \( f(1) = 2 \):
\[
f(0) = a \cdot b^0 = a \cdot 1 = a = 1
\]
\[
f(1) = 1 \cdot b^1 = b = 2
\]
Thus, \( b = 2 \).
3. **Construct the Function:**
Since we have found \( a = 1 \) and \( b = 2 \), the exponential function that represents the graph is:
\[
f(x) = 1 \cdot 2^x = 2^x
\]
**Explanation of Key Points:**
- The y-value increases very gradually for negative x-values (close to 0 but non-negative).
- For positive x-values, the y-value grows at an exponential rate, confirming the exponential behavior as described by \(
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