To determine the equation for the horizontal asymptote of an exponential function, we need to understand the general form of an exponential function and how shifts affect this form.
An exponential function typically has the form:
[tex]\[ y = a \cdot b^x + c \][/tex]
where:
- [tex]\(a\)[/tex] is the initial value (when [tex]\(x = 0\)[/tex]),
- [tex]\(b\)[/tex] is the base of the exponential function, indicating the growth or decay factor,
- [tex]\(c\)[/tex] is the vertical shift, which directly impacts the horizontal asymptote.
Given that [tex]\(h(x)\)[/tex] decays at a rate of [tex]\(65\%\)[/tex], it implies that the decay factor [tex]\(b\)[/tex] is [tex]\(1 - 0.65 = 0.35\)[/tex].
The function passes through the point [tex]\((0, 6)\)[/tex]. When [tex]\(x = 0\)[/tex],
[tex]\[ h(0) = a \cdot 0.35^0 + c = a + c \][/tex]
Since [tex]\(h(0) = 6\)[/tex], we have:
[tex]\[ a + c = 6 \][/tex]
We are also informed that the function is shifted up by 4 units. Thus, [tex]\(c = 4\)[/tex].
Considering this vertical shift upwards, the horizontal asymptote of the exponential function is determined by the value of [tex]\(c\)[/tex]. The horizontal asymptote is represented by the equation:
[tex]\[ y = c \][/tex]
Substituting the value [tex]\(c = 4\)[/tex]:
Therefore, the equation for the horizontal asymptote is:
[tex]\[ y = 4 \][/tex]
Thus, the correct multiple choice answer is:
[tex]\[ y = 4 \][/tex]
