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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![**Title: Determining the Function for the Given Graph**
**Introduction:**
This section provides instructions on identifying the equation of a depicted graph, emphasizing key characteristics and the behavior of the graph. Understanding these elements is crucial for formulating the correct mathematical equation.
**Graph Analysis:**
The graph presented is a classic exponential function. To discern its specific equation, examine the following attributes:
1. **Axis and Quadrants:**
- The graph is displayed on a Cartesian coordinate system with both x-axis and y-axis extending from -5 to 5.
2. **Growth or Decay:**
- It is essential to determine whether the graph represents exponential growth or decay. In this case, as the x-values increase, the y-values rise sharply, indicating exponential growth.
3. **Function Behavior:**
- The y-values become negatively large as x approaches zero from the negative side. Conversely, as x passes zero and continues to grow, so do the y-values, which increase quite rapidly.
4. **Base of the Exponential Function:**
- The rate of rapid growth suggests that the base of the exponent is greater than 1.
**General Form of Exponential Functions:**
The general form for an exponential function is:
\[ f(x) = a \cdot b^x \]
where:
- \(a\) is a constant.
- \(b\) is the base of the exponential function.
**Specific Equation Derivation:**
From the graph, it's visible that at \( x = 0 \), \( y \approx -5 \). This suggests that the constant \( a \) could be -5.
Given the rapid increase as \( x \) moves from negative to positive and considering typical exponential functions, we might infer:
\[ f(x) = -5 \cdot e^x \]
or another fitting constant for the increasing nature.
**Graph Interpretation:**
- **X-axis (Horizontal Axis):** Scaled in intervals of 1 from -5 to 5.
- **Y-axis (Vertical Axis):** Scaled similarly from -8 to 8.
- **Curve behavior:** The plot starts from the left, closely hugging the negative axis, then exponentially surges upwards past zero.
**Function Verification:**
To verify the derived function:
- Plug in values of \( x \) and compare the resulting \( y \).
- Confirm through points such as \( x = 0 \), ensuring they comply](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58deed7b-e5cf-455c-8dcf-5c14f5ab5a29%2Fc88ed328-1635-460e-a7b5-72a1b0dddcc6%2Fgvd0lo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determining the Function for the Given Graph**
**Introduction:**
This section provides instructions on identifying the equation of a depicted graph, emphasizing key characteristics and the behavior of the graph. Understanding these elements is crucial for formulating the correct mathematical equation.
**Graph Analysis:**
The graph presented is a classic exponential function. To discern its specific equation, examine the following attributes:
1. **Axis and Quadrants:**
- The graph is displayed on a Cartesian coordinate system with both x-axis and y-axis extending from -5 to 5.
2. **Growth or Decay:**
- It is essential to determine whether the graph represents exponential growth or decay. In this case, as the x-values increase, the y-values rise sharply, indicating exponential growth.
3. **Function Behavior:**
- The y-values become negatively large as x approaches zero from the negative side. Conversely, as x passes zero and continues to grow, so do the y-values, which increase quite rapidly.
4. **Base of the Exponential Function:**
- The rate of rapid growth suggests that the base of the exponent is greater than 1.
**General Form of Exponential Functions:**
The general form for an exponential function is:
\[ f(x) = a \cdot b^x \]
where:
- \(a\) is a constant.
- \(b\) is the base of the exponential function.
**Specific Equation Derivation:**
From the graph, it's visible that at \( x = 0 \), \( y \approx -5 \). This suggests that the constant \( a \) could be -5.
Given the rapid increase as \( x \) moves from negative to positive and considering typical exponential functions, we might infer:
\[ f(x) = -5 \cdot e^x \]
or another fitting constant for the increasing nature.
**Graph Interpretation:**
- **X-axis (Horizontal Axis):** Scaled in intervals of 1 from -5 to 5.
- **Y-axis (Vertical Axis):** Scaled similarly from -8 to 8.
- **Curve behavior:** The plot starts from the left, closely hugging the negative axis, then exponentially surges upwards past zero.
**Function Verification:**
To verify the derived function:
- Plug in values of \( x \) and compare the resulting \( y \).
- Confirm through points such as \( x = 0 \), ensuring they comply
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