To determine the horizontal asymptote of the given exponential function, let's consider the key components of the problem:
1. Exponential Function Characteristics:
- The function [tex]\( h(x) \)[/tex] decays at a rate of 65%. This information usually pertains to the base of the exponential function in the form [tex]\( h(x) = a \cdot b^x \)[/tex], where [tex]\( b = 0.65 \)[/tex] because the function decays at this rate.
2. Ordered Pair:
- The function passes through the point (0, 5). This indicates that when [tex]\( x = 0 \)[/tex], [tex]\( h(x) = 5 \)[/tex]. Thus, [tex]\( a = 5 \)[/tex], making the original function [tex]\( h(x) = 5 \cdot 0.65^x \)[/tex].
3. Vertical Shift:
- The function is shifted up by 3 units. When a function [tex]\( h(x) \)[/tex] is shifted vertically, the equation becomes [tex]\( g(x) = h(x) + k \)[/tex], where [tex]\( k \)[/tex] is the vertical shift. In this case, [tex]\( k = 3 \)[/tex], so the shifted function is [tex]\( g(x) = 5 \cdot 0.65^x + 3 \)[/tex].
4. Horizontal Asymptote:
- For an exponential function [tex]\( h(x) \)[/tex], regardless of whether it grows or decays, as [tex]\( x \to \infty \)[/tex] (positive or negative), the term [tex]\( 5 \cdot 0.65^x \)[/tex] tends towards 0 because [tex]\( 0 < 0.65 < 1 \)[/tex].
Therefore, the horizontal asymptote of the function [tex]\( g(x) = 5 \cdot 0.65^x + 3 \)[/tex] will be determined by the constant term added to the function. As [tex]\( x \)[/tex] approaches infinity, [tex]\( 5 \cdot 0.65^x \)[/tex] decays to 0, and the function approaches the value of the constant term.
Given the shift up by 3 units, the horizontal asymptote is:
[tex]\[ y = 3 \][/tex]
So, the correct answer is:
[tex]\[ y = 3 \][/tex]
