To determine the asymptote and the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) = 6 \cdot (0.5)^x + 2 \)[/tex], follow these steps:
### Finding the Asymptote
1. An asymptote is a line that the graph of a function approaches but never touches.
2. For the function [tex]\( f(x) = 6 \cdot (0.5)^x + 2 \)[/tex], observe that as [tex]\( x \)[/tex] approaches infinity, the term [tex]\( (0.5)^x \)[/tex] approaches 0 because raising 0.5 to a higher power makes it smaller.
3. Therefore, the function simplifies to [tex]\( f(x) \approx 2 \)[/tex] as [tex]\( x \)[/tex] approaches infinity.
Hence, the horizontal asymptote of the function is [tex]\( y = 2 \)[/tex].
### Finding the [tex]\( y \)[/tex]-Intercept
1. The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(0) = 6 \cdot (0.5)^0 + 2
\][/tex]
3. Simplify the equation:
[tex]\[
(0.5)^0 = 1
\][/tex]
So,
[tex]\[
f(0) = 6 \cdot 1 + 2 = 8
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is at the point [tex]\( (0, 8) \)[/tex].
### Conclusion
Based on the calculations:
- The asymptote is [tex]\( y = 2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0, 8) \)[/tex].
So the correct answer is:
A. asymptote: [tex]\( y = 2 \)[/tex], [tex]\( y \)[/tex]-intercept: [tex]\( (0, 8) \)[/tex]
