(25)*
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Graph the following exponential function. Show work in how you find the y-intercept, talk about the end behavior, and state if it is a growth or decay function.
![The image contains the following mathematical expression:
\[ y = -\frac{1}{2} (25)^{x-1} + 2 \]
In this equation:
1. **\( y \)** is the dependent variable.
2. **\( -\frac{1}{2} \)** is a constant coefficient.
3. **\( 25 \)** is the base of the exponential term.
4. **\( x-1 \)** is the exponent of the base \( 25 \).
5. **\( +2 \)** is a constant term added to the entire expression.
This is an exponential function where the variable \( x \) is in the exponent. The function will exhibit exponential growth or decay, depending on the value of \( x \). The coefficient \( -\frac{1}{2} \) will affect the amplitude and the sign of the function, making it reflect across the x-axis if needed. The \( +2 \) term will shift the entire graph vertically by 2 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeea96f9-d9e2-4e41-9df9-d54882930b66%2F800f3799-ad48-41f5-b2a1-f5e22c54a1b1%2F80dyctl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the following mathematical expression:
\[ y = -\frac{1}{2} (25)^{x-1} + 2 \]
In this equation:
1. **\( y \)** is the dependent variable.
2. **\( -\frac{1}{2} \)** is a constant coefficient.
3. **\( 25 \)** is the base of the exponential term.
4. **\( x-1 \)** is the exponent of the base \( 25 \).
5. **\( +2 \)** is a constant term added to the entire expression.
This is an exponential function where the variable \( x \) is in the exponent. The function will exhibit exponential growth or decay, depending on the value of \( x \). The coefficient \( -\frac{1}{2} \) will affect the amplitude and the sign of the function, making it reflect across the x-axis if needed. The \( +2 \) term will shift the entire graph vertically by 2 units.
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