Sketch the graph of an exponential function that has the following properties:
• domain (r E R)
• range {y E R, y > 0)
• y-intercept 3
• horizontal asymptote y = 0
• decreasing function
b) Is it possible to sketch the graph of another exponential function with the same properties? Explain.

Sketch the graph of an exponential function that has the following properties:
• domain (r E R)
• range {y E R, y > 0)
• y-intercept 3
• horizontal asymptote y = 0
• decreasing function
b) Is it possible to sketch the graph of another exponential function with the same properties? Explain.

Sketch the graph of an exponential function that has the following properties:
• domain (r E R)
• range {y E R, y > 0)
• y-intercept 3
• horizontal asymptote y = 0
• decreasing function
b) Is it possible to sketch the graph of another exponential function with the same properties? Explain.

Sketch the graph of an exponential function that has the following properties:
• domain (r E R)
• range {y E R, y > 0)
• y-intercept 3
• horizontal asymptote y = 0
• decreasing function
b) Is it possible to sketch the graph of another exponential function with the same properties? Explain.

Let's consider the properties of the given exponential function:

### Properties:
1. **Domain**: \( \mathbb{R} \) (all real numbers).
2. **Range**: \( y > 0 \) (the function's values are always positive).
3. **Y-intercept**: 3 (the function crosses the y-axis at \( y = 3 \)).
4. **Horizontal asymptote**: \( y = 0 \) (the function approaches zero but never touches or crosses it as \( x \to \infty \)).
5. **Decreasing function**: The function is decreasing, meaning the function's value gets smaller as \( x \) increases.

### General Form:
The general form of an exponential function is:

\[
y = ab^x
\]

For the function to be decreasing, \( 0 < b < 1 \). The y-intercept is 3, meaning when \( x = 0 \), \( y = 3 \), so \( a = 3 \). Therefore, the function has the form:

\[
y = 3b^x
\]

where \( 0 < b < 1 \).

### Graph Sketch:
- The graph will pass through the point \( (0, 3) \).
- As \( x \to \infty \), the function will approach \( y = 0 \) but never touch the x-axis (the horizontal asymptote).
- As \( x \to -\infty \), the function will increase rapidly.
- The graph is decreasing since \( 0 < b < 1 \).

I can generate a sketch for this.

### Part (b):
Is it possible to sketch another exponential function with the same properties?

No, it is not possible to sketch another function with the same properties. The key constraints, such as the domain \( \mathbb{R} \), range \( y > 0 \), the horizontal asymptote at \( y = 0 \), and the decreasing behavior, are all uniquely described by the function \( y = 3b^x \) with \( 0 < b < 1 \). Changing any of these would violate one or more of the given conditions.

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Would you like me to generate a visual sketch of this graph? Also, do you want more details on any part?

Here are five related questions to expand your understanding:
1. How does changing the base \( b \) affect the steepness of the graph?
2. Can an exponential function have a horizontal asymptote at any value other than zero?
3. What is the behavior of the function as \( x \to -\infty \)?
4. How does the graph of a decreasing exponential function compare to that of an increasing one?
5. How would you transform this function to move the asymptote to a different y-value?

**Tip**: To identify the behavior of an exponential function, pay close attention to the base \( b \) and the asymptote—it tells you a lot about the growth or decay pattern.

Sketch the graph of an exponential function that has the following properties: • domain (r E R) • range {y E R, y > 0) • y-intercept 3 • horizontal asymptote y = 0 • decreasing function b) Is it possible to sketch the graph of another exponential function with the same properties? Explain.

Learn how to sketch the graph of a decreasing exponential function with a y-intercept of 3 and horizontal asymptote at y = 0. This problem covers key concepts like domain, range, and properties of exponential functions. Suitable for high school students in Grades 9-12.

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