**Instruction for Educational Website****Task 16: Graphing an Exponential Function****Objective:** Graph the function $ f(x) = \frac{1}{4}(2)^x - 3 $.**Step-by-step Explanation:**1. **Function Analysis:** - The given function is $ f(x) = \frac{1}{4}(2)^x - 3 $. - It is an exponential function where the base is 2, and there's a vertical transformation. - The $\frac{1}{4}$ is a vertical compression, and $-3$ is a vertical shift downwards by 3 units.2. **Key Features:** - **Asymptote:** The horizontal asymptote is at $ y = -3 $ due to the vertical shift. - **Y-intercept:** Calculate $ f(0) = \frac{1}{4}(2)^0 - 3 = \frac{1}{4} - 3 = -\frac{11}{4} $. - **Growth/Decay:** Since the base $2 > 1$, it is a growth function.3. **Plotting the Graph:** - **Step 1:** Plot the asymptote line at $ y = -3 $. - **Step 2:** Identify and plot the y-intercept $ (0, -\frac{11}{4}) $. - **Step 3:** Choose additional x-values to calculate respective y-values for accurate plotting. - For example, calculate $ f(1), f(2), $ and $ f(-1) $, etc. - **Step 4:** Sketch the curve passing through these points, showing exponential growth approaching the asymptote as $ x \to -\infty $ and rising to the right.4. **Graph Behavior:** - The graph rises exponentially to the right due to the positive base. - It asymptotically approaches $ y = -3 $ as $ x \to -\infty $.This completes the task of graphing the function $ f(x) = \frac{1}{4}(2)^x - 3 $.

The given function is , fx=142x-3

Answered: 16. Graph the function f(x)=- (2)*– 3

Math

Algebra

16. Graph the function f(x)=- (2)*– 3

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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Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.

Question

**Instruction for Educational Website**

**Task 16: Graphing an Exponential Function**

**Objective:** Graph the function $ f(x) = \frac{1}{4}(2)^x - 3 $.

**Step-by-step Explanation:**

1. **Function Analysis:**
- The given function is $ f(x) = \frac{1}{4}(2)^x - 3 $.
- It is an exponential function where the base is 2, and there's a vertical transformation.
- The $\frac{1}{4}$ is a vertical compression, and $-3$ is a vertical shift downwards by 3 units.

2. **Key Features:**
- **Asymptote:** The horizontal asymptote is at $ y = -3 $ due to the vertical shift.
- **Y-intercept:** Calculate $ f(0) = \frac{1}{4}(2)^0 - 3 = \frac{1}{4} - 3 = -\frac{11}{4} $.
- **Growth/Decay:** Since the base $2 > 1$, it is a growth function.

3. **Plotting the Graph:**
- **Step 1:** Plot the asymptote line at $ y = -3 $.
- **Step 2:** Identify and plot the y-intercept $ (0, -\frac{11}{4}) $.
- **Step 3:** Choose additional x-values to calculate respective y-values for accurate plotting.
- For example, calculate $ f(1), f(2), $ and $ f(-1) $, etc.
- **Step 4:** Sketch the curve passing through these points, showing exponential growth approaching the asymptote as $ x \to -\infty $ and rising to the right.

4. **Graph Behavior:**
- The graph rises exponentially to the right due to the positive base.
- It asymptotically approaches $ y = -3 $ as $ x \to -\infty $.

This completes the task of graphing the function $ f(x) = \frac{1}{4}(2)^x - 3 $.

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