## Transcription of Exponential Function Analysis**4. Consider the exponential function $ y = -3 - 2^{-x} + 5 $**### a. Algebraically determine the exact value of the y-intercept.To find the y-intercept, substitute $ x = 0 $ into the equation:\[ y = -3 - 2^{-(0)} + 5 \]\[ y = -3 - 1 + 5 \]\[ y = 4 \]Thus, the y-intercept is $ (0, 4) $.### b. Complete the table. Show transformations for determining critical points.| | ||--------------------------|----------------------------------|| **Parent Function** | $ 2^x $ || | $ (0,1) $ || **Critical Point** | $ (5, -2) $ || **Intercepts** | x-intercept: none || | y-intercept: $ (0, 4) $ || **Asymptotes** | $ y = -3 $ || | x = none || **Domain** | $ \mathbb{R} $ || **Interval $ f(x) $ is decreasing** | $ \mathbb{R} $ || **End Behavior** | As $ x \to -\infty, y \to -3 $ |### c. Sketch the graph showing the critical point, all intercepts, and asymptotes.- The graph is not shown in detail due to the medium; however, it illustrates an exponential curve, reflecting the described function characteristics and manipulation. - The critical point is at $ (5, -2) $.- The horizontal asymptote at $ y = -3 $ is indicated by a horizontal dashed line.- There is no x-intercept present.- The curve approaches the asymptote as $ x $ decreases toward negative infinity.

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4. Consider the exponential function y = -3-2-*+5 a. Algebraically determine the exact value of the y-intercept.

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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## Transcription of Exponential Function Analysis

**4. Consider the exponential function $ y = -3 - 2^{-x} + 5 $**

### a. Algebraically determine the exact value of the y-intercept.

To find the y-intercept, substitute $ x = 0 $ into the equation:

\[ y = -3 - 2^{-(0)} + 5 \]

\[ y = -3 - 1 + 5 \]

\[ y = 4 \]

Thus, the y-intercept is $ (0, 4) $.

### b. Complete the table. Show transformations for determining critical points.

| | |
|--------------------------|----------------------------------|
| **Parent Function** | $ 2^x $ |
| | $ (0,1) $ |
| **Critical Point** | $ (5, -2) $ |
| **Intercepts** | x-intercept: none |
| | y-intercept: $ (0, 4) $ |
| **Asymptotes** | $ y = -3 $ |
| | x = none |
| **Domain** | $ \mathbb{R} $ |
| **Interval $ f(x) $ is decreasing** | $ \mathbb{R} $ |
| **End Behavior** | As $ x \to -\infty, y \to -3 $ |

### c. Sketch the graph showing the critical point, all intercepts, and asymptotes.

- The graph is not shown in detail due to the medium; however, it illustrates an exponential curve, reflecting the described function characteristics and manipulation.
- The critical point is at $ (5, -2) $.
- The horizontal asymptote at $ y = -3 $ is indicated by a horizontal dashed line.
- There is no x-intercept present.
- The curve approaches the asymptote as $ x $ decreases toward negative infinity.

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