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An exponential function, f(x), passes through (4,-2) and (5,10) and has a horizontal asymptote at y=-8. What is an equation of f(x) ?
f(x)=◻

An exponential function, $f(x)$ , passes through $(4,-2)$ and $(5,10)$ and has a horizontal asymptote at $y=-8$ . What is an equation of $f(x)$ ? $\newline$ $f(x)=\square$

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Write equations of cosine functions using properties

Full solution

Q. An exponential function,

f(x)

, passes through

(4,-2)

and

(5,10)

and has a horizontal asymptote at

y=-8

. What is an equation of

f(x)

?

\newline

f(x)=\square

Identify Asymptote: An exponential function with a horizontal asymptote at $y = -8$ can be written in the form $f(x) = a \cdot b^x + c$ , where $c$ is the horizontal asymptote. So, $c = -8$ .
Find Values of a and b: We have: $f(x) = a \cdot b^x - 8$ . Now we need to find the values of $a$ and $b$ using the points $(4, -2)$ and $(5, 10)$ .
Equation from Point $(4, -2)$ : Using the point $(4, -2)$ , we get the equation $-2 = a \cdot b^4 - 8$ . Adding $8$ to both sides gives us $a \cdot b^4 = 6$ .
Equation from Point $(5, 10)$ : Using the point $(5, 10)$ , we get the equation $10 = a \cdot b^5 - 8$ . Adding $8$ to both sides gives us $a \cdot b^5 = 18$ .
Eliminate $a$ to Solve for $b$ : Now we have two equations: $a \cdot b^4 = 6$ and $a \cdot b^5 = 18$ . We can divide the second equation by the first to eliminate $a$ and solve for $b$ . $\newline$ $\frac{a \cdot b^5}{a \cdot b^4} = \frac{18}{6}$ $\newline$ $b = 3$
Substitute $b$ to Find $a$ : Now that we have $b$ , we can substitute it back into one of the equations to find $a$ . Using $a \cdot b^4 = 6$ and $b = 3$ , we get: $\newline$ $a \cdot 3^4 = 6$ $\newline$ $a \cdot 81 = 6$ $\newline$ $a = \frac{6}{81}$ $\newline$ $a = \frac{1}{13.5}$
Final Exponential Function: Substituting $a = \frac{1}{13.5}$ and $b = 3$ into $f(x) = a \cdot b^x - 8$ gives us the final equation $f(x) = \left(\frac{1}{13.5}\right) \cdot 3^x - 8$ .

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