Question

An exponential function passes through the points 1,18 and 3,48 and has an asymptote of y=8 write the exponential function

Answer 1

Answer:

$f(x)=5\cdot 2^x + 8$

Step-by-step explanation:

To find the equation for the exponential function passing through the points (1, 18) and (3, 48), and with an asymptote at y = 8, we can use the general form of an exponential function:

$\boxed{\begin{array}{l}\underline{\textsf{General form of an Exponential Function}}\\\\ y=ab^x+c \\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ a is the initial value ( y -intercept).}\\ \phantom{ww}\bullet\;\textsf{ b is the base (growth/decay factor) in decimal form.}\\\phantom{ww}\bullet\;\textsf{ y=c is the horizontal asymptote.}\\ \end{array}}$

In this case the horizontal asymptote is y = 8, so c = 8:

$y = ab^x + 8$

Substitute the given points into the equation to form two equations:

$\underline{\textsf{Equation 1}}\\\\ab^1+8=18\\\\ab=10$

$\underline{\textsf{Equation 2}}\\\\ab^3+8=48\\\\ab^3=40$

Divide the second equation by the first equation to eliminate a, then solve for b:

$\dfrac{ab^3}{ab}=\dfrac{40}{10}\\\\\\b^2=4\\\\\\\sqrt{b^2}=\sqrt{4}\\\\\\b=2$

We always take the positive square root of b because when the base of an exponential function is negative, the function can result in complex values, leading to non-real solutions. As a consequence, an exponential function with a negative base does not have the ability to exhibit consistent growth or decay.

Now, substitute b = 2 into the equation ab = 10 and solve for a:

$a( 2)=10\\\\\\a=\dfrac{10}{ 2}\\\\\\a= 5$

Substitute the values of a, b, and c back into the exponential function:

$y=5\cdot 2^x + 8$

Therefore, the exponential function that passes through the points (1, 18) and (3, 48) and has an asymptote of y = 8 is:

$\Large\boxed{\boxed{f(x)=5\cdot 2^x + 8}}$