Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,4)$ and $(5,128)$ . $\newline$ Answer:

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,4)

and

(5,128)

\newline

Answer:

Find 'a' using first point: Given points: $(0, 4)$ and $(5, 128)$ . We will use these points to find the values of 'a' and 'b' in the exponential function $y = ab^{x}$ . $\newline$ First, we will substitute the first point $(0, 4)$ into the equation to find 'a'. $\newline$ $y = ab^{x}$ $\newline$ $4 = ab^{0}$ $\newline$ Since anything raised to the power of $0$ is $1$ , we have: $\newline$ $4 = a \times 1$ $\newline$ $4 = a$ $\newline$ So, 'a' is $(5, 128)$ $0$ .
Find 'b' using second point: Now that we have 'a', we will use the second point $(5, 128)$ to find 'b'. $\newline$ Substitute 'a' and the second point into the equation: $\newline$ $128 = 4b^{5}$ $\newline$ To solve for 'b', we divide both sides by $4$ : $\newline$ $\frac{128}{4} = b^{5}$ $\newline$ $32 = b^{5}$ $\newline$ Now we need to find the fifth root of $32$ to solve for 'b': $\newline$ $b = 32^{\frac{1}{5}}$ $\newline$ $b = 2$ $\newline$ So, 'b' is $2$ .
Write exponential function: We have found $a$ to be $4$ and $b$ to be $2$ . Now we can write the exponential function using these values: $\newline$ $y = a b^{x}$ $\newline$ $y = 4 \times 2^{x}$ $\newline$ This is the exponential function that passes through the points $(0, 4)$ and $(5, 128)$ .