realize 5-3: MothxL. for Schoot: Additional Practioe (LMS groded) Explain the key features of the exponential function $$ \mathbf{y}=\mathbf{a} \cdot \mathbf{b}^{x} $$, including the asymptote, key points on the graph, domain, and range. The graph has an asymptote at $$ \square $$ and $$ \square $$ at $$ (0, a) $$. The domain is $$ \square $$ and the range is

Question

UpStudy AI · Accepted Answer

Certainly! Let's delve into the key features of the exponential function $$ \mathbf{y} = \mathbf{a} \cdot \mathbf{b}^{x} $$. Understanding these features will help you graph the function accurately and comprehend its behavior.

### 1. **Asymptote**
An **asymptote** is a line that the graph of a function approaches but never touches or crosses. For the exponential function $$ y = a \cdot b^{x} $$:

- **Horizontal Asymptote:** The graph of an exponential function always has a horizontal asymptote. Specifically:
  
  $$
  y = 0
  $$
  
  This means as $$ x $$ approaches positive or negative infinity, $$ y $$ gets closer and closer to 0 but never actually reaches it.

### 2. **Key Points on the Graph**

- **Initial Point ($$ x = 0 $$):**
  
  When $$ x = 0 $$,
  
  $$
  y = a \cdot b^{0} = a \cdot 1 = a
  $$
  
  So, the graph always passes through the point:
  
  $$
  (0, \mathbf{a})
  $$
  
- **Another Key Point ($$ x = 1 $$):**
  
  When $$ x = 1 $$,
  
  $$
  y = a \cdot b^{1} = a \cdot b
  $$
  
  This gives us the point:
  
  $$
  (1, \mathbf{a \cdot b})
  $$
  
- **Additional Points:**
  
  By choosing different values of $$ x $$, you can plot more points to get a clearer picture of the graph's shape.

### 3. **Domain and Range**

- **Domain:** The **domain** of a function is the set of all possible input values (values of $$ x $$).

For $$ y = a \cdot b^{x} $$:
  
  $$
  \text{Domain: } (-\infty, \infty)
  $$
  
  This means $$ x $$ can be any real number.

- **Range:** The **range** of a function is the set of all possible output values (values of $$ y $$).

For $$ y = a \cdot b^{x} $$, assuming $$ a > 0 $$ and $$ b > 0 $$:
  
  $$
  \text{Range: } (0, \infty)
  $$
  
  This means $$ y $$ is always positive and can take any positive real value.

### 4. **Filling in the Blanks**

Based on the above explanations, we can complete the provided sentences:

- **Asymptote and Key Point:**

> The graph has an asymptote at $$ \mathbf{y = 0} $$ and a key point at $$ (0, \mathbf{a}) $$.

- **Domain and Range:**

> The domain is $$ \mathbf{(-\infty, \infty)} $$ and the range is $$ \mathbf{(0, \infty)} $$.

### 5. **Summary of Key Features**

| Feature          | Description                                  |
|------------------|----------------------------------------------|
| **Asymptote**    | Horizontal line at $$ y = 0 $$               |
| **Key Point 1**  | $$ (0, a) $$                                  |
| **Domain**       | All real numbers $$ (-\infty, \infty) $$     |
| **Range**        | All positive real numbers $$ (0, \infty) $$  |

### 6. **Graphical Representation**

Here's a basic sketch to visualize these features:

```
y
|
|                 *
|               *
|             *
|           *
|_________*____________ x
        (0, a)
```

- The dashed horizontal line represents the asymptote at $$ y = 0 $$.
- The point $$ (0, a) $$ is where the graph crosses the y-axis.

### 7. **Example**

Let's consider an example to solidify these concepts.

**Example:** Graph $$ y = 2 \cdot 3^{x} $$

- **Asymptote:** $$ y = 0 $$
- **Key Point:** $$ (0, 2) $$
- **Domain:** $$ (-\infty, \infty) $$
- **Range:** $$ (0, \infty) $$

**Points to Plot:**

| $$ x $$ | $$ y = 2 \cdot 3^{x} $$ |
|---------|------------------------|
| -1      | $$ 2 \cdot 3^{-1} = \frac{2}{3} $$ |
| 0       | $$ 2 \cdot 3^{0} = 2 $$             |
| 1       | $$ 2 \cdot 3^{1} = 6 $$             |
| 2       | $$ 2 \cdot 3^{2} = 18 $$            |

Plot these points and draw the curve approaching the asymptote $$ y = 0 $$.

---

By understanding these key features, you can effectively analyze and graph any exponential function of the form $$ y = a \cdot b^{x} $$.
The exponential function $$ y = a \cdot b^{x} $$ has a horizontal asymptote at $$ y = 0 $$ and passes through the point $$ (0, a) $$. Its domain is all real numbers $$ (-\infty, \infty) $$ and its range is all positive real numbers $$ (0, \infty) $$.

Real Tutor Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions