Identify the graph for the equation. $$ y=4\left(\frac{1}{2}\right)^{x} $$

Question

UpStudy AI · Accepted Answer

The equation you've provided is an exponential function:

$$
y = 4\left(\frac{1}{2}\right)^{x}
$$

Let's break down the components to understand and identify its graph:

### 1. **Base and Exponent**
- **Base ($$ \frac{1}{2} $$)**: Since the base is between 0 and 1 ($$ 0 < \frac{1}{2} < 1 $$), this indicates that the function represents **exponential decay**. This means as $$ x $$ increases, $$ y $$ decreases.
  
### 2. **Vertical Stretch**
- **Coefficient (4)**: The multiplier 4 stretches the graph vertically by a factor of 4. This affects the **y-intercept** of the graph.

### 3. **Y-Intercept**
- To find the y-intercept, set $$ x = 0 $$:
  $$
  y = 4\left(\frac{1}{2}\right)^{0} = 4 \times 1 = 4
  $$
  So, the graph passes through the point **(0, 4)**.

### 4. **Horizontal Asymptote**
- For exponential functions of the form $$ y = A \cdot b^x $$, where $$ 0 < b < 1 $$, the horizontal asymptote is:
  $$
  y = 0
  $$
  This means the graph approaches the x-axis but never touches it as $$ x $$ increases.

### 5. **Behavior**
- **As $$ x \to \infty $$**: 
  - $$ y \to 0 $$ (approaches the horizontal asymptote).
- **As $$ x \to -\infty $$**:
  - $$ y \to \infty $$ (the function increases without bound).

### 6. **Graph Characteristics**
- **Decreasing Curve**: Since the base is $$ \frac{1}{2} $$, the graph will decrease from left to right.
- **Y-Intercept at (0, 4)**: Confirms the starting point of the graph.
- **Approaches $$ y = 0 $$ as $$ x $$ increases**: Represents the horizontal asymptote.

### **Graph Summary**

The graph of $$ y = 4\left(\frac{1}{2}\right)^{x} $$ is a **decreasing exponential curve** that:
- Passes through the point **(0, 4)**.
- Has a **horizontal asymptote** at $$ y = 0 $$.
- Approaches infinity as $$ x $$ becomes very negative.
- Approaches zero as $$ x $$ becomes very positive.

Here's a visual representation of the graph:

![Graph of y = 4*(1/2)^x](https://i.imgur.com/YourGraphImageLink.png) *(Please replace with an actual graph if needed.)*

### **Key Points to Identify the Graph**
- **Exponential Decay**: Decreasing as $$ x $$ increases.
- **Y-Intercept at 4**: The graph crosses the y-axis at (0, 4).
- **Horizontal Asymptote**: Approaches the x-axis but never touches it.

I hope this helps you identify and understand the graph of the given exponential function!
The graph of $$ y = 4\left(\frac{1}{2}\right)^{x} $$ is a decreasing exponential curve that passes through the point (0, 4) and approaches the x-axis as $$ x $$ increases.

Identify the graph for the equation. \[ y=4\left(\frac{1}{2}\right)^{x} \]

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