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Given an exponential function of the form f(x)=bx,
 graph the function.

Create a table of points.
Plot at least 3 points from the table, including the y-intercept (0,1).
Draw a smooth curve through the points.
State the domain, (−∞,∞), the range, (0,∞), and the horizontal asymptote, y=0.

The image you provided shows two exponential functions of the form \( f(x) = b^x \), which illustrates the key features for different values of \( b \). Let's summarize the two cases and generate the table, graphs, domain, range, and asymptotes for each case:

### 1. **Case 1: \( f(x) = b^x \), where \( b > 1 \)**
   - This is an **exponential growth** function.

   #### Table of Points:
   | \( x \)  | \( f(x) = b^x \) |
   |:-------:|:--------------:|
   |  -1     | \( \frac{1}{b} \) |
   |   0     | 1              |
   |   1     | \( b \)         |

   #### Plot:
   - For \( b > 1 \), you have three points: 
     - \( (-1, \frac{1}{b}) \)
     - \( (0, 1) \)  (the y-intercept)
     - \( (1, b) \)

   - The curve increases rapidly as \( x \) increases.

   #### Domain and Range:
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( (0, \infty) \)
   - **Horizontal Asymptote**: \( y = 0 \) (as \( x \to -\infty \), the function approaches 0).

### 2. **Case 2: \( f(x) = b^x \), where \( 0 < b < 1 \)**
   - This is an **exponential decay** function.

   #### Table of Points:
   | \( x \)  | \( f(x) = b^x \) |
   |:-------:|:--------------:|
   |  -1     | \( \frac{1}{b} \) |
   |   0     | 1              |
   |   1     | \( b \)         |

   #### Plot:
   - For \( 0 < b < 1 \), the key points are:
     - \( (-1, \frac{1}{b}) \)
     - \( (0, 1) \)
     - \( (1, b) \)

   - The curve decreases rapidly as \( x \) increases.

   #### Domain and Range:
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( (0, \infty) \)
   - **Horizontal Asymptote**: \( y = 0 \) (as \( x \to \infty \), the function approaches 0).

### Graph Summary:
- **Left graph**: Represents the exponential growth case \( b > 1 \).
- **Right graph**: Represents the exponential decay case \( 0 < b < 1 \).

Would you like me to generate a plot for any specific values of \( b \) or clarify any other details?

### Related Questions:
1. How does changing the base \( b \) affect the steepness of the exponential graph?
2. Can an exponential function ever cross the horizontal asymptote \( y = 0 \)?
3. How can you determine whether an exponential function represents growth or decay just by looking at the base \( b \)?
4. What happens to the graph when \( b = 1 \)?
5. What is the effect of negative exponents on the exponential function?

### Tip:
Always remember, for exponential functions, the y-intercept is always \( (0,1) \), regardless of the base \( b \), since \( b^0 = 1 \) for any \( b > 0 \).

Given an exponential function of the form f(x) = b^x, graph the function. Create a table of points. Plot at least 3 points from the table, including the y-intercept (0,1). Draw a smooth curve through the points. State the domain, (−∞, ∞), the range, (0, ∞), and the horizontal asymptote, y = 0.

Learn how to graph exponential functions of the form f(x) = b^x. The problem covers exponential growth (b > 1) and decay (0 < b < 1), and includes plotting points, determining the domain, range, and the horizontal asymptote y = 0. Suitable for students in Grades 9-12.

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