The function on the first attachment has the form f(x) = b(square) + k. Which of the following functions is shown on the graph? Attachment 1 - Graph Attachment 2 - Multiple choice
The function on the first attachment has the form f(x) = b(square) + k. Which of the following functions is shown on the graph? Attachment 1 - Graph Attachment 2 - Multiple choice
The function on the first attachment has the form f(x) = b(square) + k. Which of the following functions is shown on the graph? Attachment 1 - Graph Attachment 2 - Multiple choice
The function on the first attachment has the form f(x) = b(square) + k. Which of the following functions is shown on the graph?
Attachment 1 - Graph
Attachment 2 - Multiple choice
Transcribed Image Text:### Understanding Exponential Functions
The function below has the form \( f(x) = b^x + k \).
This graph depicts an exponential function, characterized by the general form \( f(x) = b^x + k \). Here's a step-by-step breakdown of the graph shown:
#### Graph Explanation:
- **Axes**:
- The horizontal axis represents the \( x \)-values, ranging from -5 to 5.
- The vertical axis represents the \( y \)-values, ranging from 0 to 14.
- **Curve Characteristics**:
- The function's curve starts near the x-axis on the left, indicating very low (possibly negative) values.
- As the function progresses rightward, it swiftly ascends after intersecting the y-axis, demonstrating exponential growth.
- **Key Points**:
- The curve passes close to the point \((0, 4)\) on the graph, indicating that \( k \) may play a role in the \( y \)-intercept.
- The graph suggests rapid growth as \( x \) increases into positive values.
#### Analyzing the Function:
- **Exponential Growth**: The upward curve represents exponential growth, typical for functions of the form \( b^x + k \) where \( b > 1 \).
- **Vertical Shift**: The constant \( k \) shifts the graph vertically. Since the intercept near \( y = 4 \), \( k \) is likely influencing this shift.
- **Base Value, \( b \)**: The base \( b \) determines the rate of growth. A larger base value \( b \) means faster growth.
#### Question for Practice:
**Which of the following functions is shown on the graph?**
- The question prompts you to identify the function from given options, likely provided in a multiple-choice format.
Analyzing the exponential function's behavior and characteristics on the graph can aid in understanding how changes in base \( b \) and \( y \)-intercept \( k \) affect the function's visualization.
Transcribed Image Text:Welcome to our Educational website's section on graph and function identification. Below you'll find a mathematical challenge designed to enhance your understanding of exponential functions. Carefully analyze the given functions and then determine which one is depicted in the accompanying graph.
### Which of the following functions is shown on the graph?
1. \( f(x) = \left( \frac{1}{5} \right)^x - 4 \)
2. \( f(x) = \left( \frac{1}{5} \right)^x + 3 \)
3. \( f(x) = \left( \frac{1}{5} \right)^x + 4 \)
4. \( f(x) = \left( \frac{1}{5} \right)^x + 5 \)
5. \( f(x) = 5^x + 4 \)
6. \( f(x) = 5^x + 3 \)
7. \( f(x) = 5^x + 5 \)
8. \( f(x) = 5^x - 4 \)
### Instructions:
- Examine the graph provided (which is not displayed here).
- Determine the characteristics of the graph, such as the y-intercept, direction of growth or decay, and any vertical shifts.
- Match these characteristics to the appropriate function listed above.
Good luck, and remember that understanding the patterns of exponential growth and decay is essential for mastering these types of functions!
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
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This graph depicts an exponential function, characterized by the general form \( f(x) = b^x + k \). Here's a step-by-step breakdown of the graph shown:
#### Graph Explanation:
- **Axes**:
- The horizontal axis represents the \( x \)-values, ranging from -5 to 5.
- The vertical axis represents the \( y \)-values, ranging from 0 to 14.
- **Curve Characteristics**:
- The function's curve starts near the x-axis on the left, indicating very low (possibly negative) values.
- As the function progresses rightward, it swiftly ascends after intersecting the y-axis, demonstrating exponential growth.
- **Key Points**:
- The curve passes close to the point \((0, 4)\) on the graph, indicating that \( k \) may play a role in the \( y \)-intercept.
- The graph suggests rapid growth as \( x \) increases into positive values.
#### Analyzing the Function:
- **Exponential Growth**: The upward curve represents exponential growth, typical for functions of the form \( b^x + k \) where \( b > 1 \).
- **Vertical Shift**: The constant \( k \) shifts the graph vertically. Since the intercept near \( y = 4 \), \( k \) is likely influencing this shift.
- **Base Value, \( b \)**: The base \( b \) determines the rate of growth. A larger base value \( b \) means faster growth.
#### Question for Practice:
**Which of the following functions is shown on the graph?**
- The question prompts you to identify the function from given options, likely provided in a multiple-choice format.
Analyzing the exponential function's behavior and characteristics on the graph can aid in understanding how changes in base \( b \) and \( y \)-intercept \( k \) affect the function's visualization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf831165-65bd-4d92-9ad7-53f462fc57d8%2F61477d12-f20f-4631-bfc6-c5ff60c50ee2%2Fiimerm1n_processed.png&w=3840&q=75)
