To solve for the [tex]\( y \)[/tex]-intercept and the equation of the asymptote of the function [tex]\( f(x) = -3^x + 2 \)[/tex], follow these steps:
### Finding the [tex]\( y \)[/tex]-Intercept:
1. The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex].
2. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = -3^x + 2 \)[/tex]:
[tex]\[
f(0) = -3^0 + 2
\][/tex]
3. Calculate the value:
[tex]\[
3^0 = 1 \implies -3^0 = -1
\][/tex]
[tex]\[
f(0) = -1 + 2 = 1
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\( y = 1 \)[/tex].
### Finding the Asymptote:
1. The asymptote of an exponential function of the form [tex]\( f(x) = -a^x + b \)[/tex] will be [tex]\( y = b \)[/tex].
2. Here, the function is [tex]\( f(x) = -3^x + 2 \)[/tex], so [tex]\( b = 2 \)[/tex].
Therefore, the asymptote is [tex]\( y = 2 \)[/tex].
### Final Answer:
- [tex]\( y \)[/tex]-intercept [tex]\( = 1 \)[/tex]
- Equation of the asymptote is [tex]\( y = 2 \)[/tex]
