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

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 \( y = a \cdot b^{x} \).