Question
3. Consider the exponential function Complete the statement below.
The y-intercept is Select. The equation of the asymptote is Select
(0-10)
y=0 x=0
(-10.0)
y=1
(0.5)
(5,0)
The y-intercept is Select. The equation of the asymptote is Select
(0-10)
y=0 x=0
(-10.0)
y=1
(0.5)
(5,0)
Answers
GPT-4o mini
To analyze the exponential function, let's consider it in a general form, such as \( y = a \cdot b^x \).
1. **Y-Intercept:** The y-intercept of an exponential function occurs when \( x = 0 \). So, we can find the y-intercept by evaluating the function at \( x = 0 \):
\[
y = a \cdot b^0 = a
\]
Thus, the y-intercept is simply \( (0, a) \).
2. **Asymptote:** The horizontal asymptote of an exponential function is usually \( y = 0 \) for functions of the form \( y = a \cdot b^x \) (where \( a \) is a constant and \( b \) is a positive constant). This means that as \( x \) approaches negative infinity, \( y \) approaches 0.
Based on your options, it seems you're looking for specific selections:
- **The y-intercept is**: It could be any of the values provided (like (0, 1), (0, 5), etc.), but without more context about the specific function, we cannot determine it definitively.
- **The equation of the asymptote is**: The answer here is \( y = 0 \).
So, to complete the statement:
- The y-intercept is **Select one of the y-intercept values you have**.
- The equation of the asymptote is **y = 0**.
1. **Y-Intercept:** The y-intercept of an exponential function occurs when \( x = 0 \). So, we can find the y-intercept by evaluating the function at \( x = 0 \):
\[
y = a \cdot b^0 = a
\]
Thus, the y-intercept is simply \( (0, a) \).
2. **Asymptote:** The horizontal asymptote of an exponential function is usually \( y = 0 \) for functions of the form \( y = a \cdot b^x \) (where \( a \) is a constant and \( b \) is a positive constant). This means that as \( x \) approaches negative infinity, \( y \) approaches 0.
Based on your options, it seems you're looking for specific selections:
- **The y-intercept is**: It could be any of the values provided (like (0, 1), (0, 5), etc.), but without more context about the specific function, we cannot determine it definitively.
- **The equation of the asymptote is**: The answer here is \( y = 0 \).
So, to complete the statement:
- The y-intercept is **Select one of the y-intercept values you have**.
- The equation of the asymptote is **y = 0**.