Question
realize
5-3: MothxL. for Schoot: Additional Practioe (LMS groded)
Explain the key features of the exponential function
5-3: MothxL. for Schoot: Additional Practioe (LMS groded)
Ask by King Chadwick. in the United States
Jan 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The exponential function
Solution
Certainly! Let’s delve into the key features of the exponential function
1. Asymptote
An asymptote is a line that the graph of a function approaches but never touches or crosses. For the exponential function
-
Horizontal Asymptote: The graph of an exponential function always has a horizontal asymptote. Specifically:This means as
approaches positive or negative infinity, gets closer and closer to 0 but never actually reaches it.
2. Key Points on the Graph
-
Initial Point (
): When, So, the graph always passes through the point: -
Another Key Point (
): When, This gives us the point: -
Additional Points:By choosing different values of
, 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
). For: This meanscan be any real number. -
Range: The range of a function is the set of all possible output values (values of
). For, assuming and : This meansis 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
and a key point at . -
Domain and Range:The domain is
and the range is .
5. Summary of Key Features
| Feature | Description |
|---|---|
| Asymptote | Horizontal line at
|
| Key Point 1 |
|
| Domain | All real numbers
|
| Range | All positive real numbers
|
6. Graphical Representation
Here’s a basic sketch to visualize these features:
y
|
| *
| *
| *
| *
|_________*____________ x
(0, a)
- The dashed horizontal line represents the asymptote at
. - The point
is where the graph crosses the y-axis.
7. Example
Let’s consider an example to solidify these concepts.
Example: Graph
- Asymptote:
- Key Point:
- Domain:
- Range:
Points to Plot:
|
|
|
|---|---|
| -1 |
|
| 0 |
|
| 1 |
|
| 2 |
|
Plot these points and draw the curve approaching the asymptote
By understanding these key features, you can effectively analyze and graph any exponential function of the form
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
The exponential function
Now let’s talk about its domain and range! The domain of an exponential function is all real numbers (
