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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,15) and
(4,1215).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,15)$ and $(4,1215)$ . $\newline$ Answer:

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Convert between exponential and logarithmic form

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,15)

and

(4,1215)

.

\newline

Answer:

Find a Value of a: We need to find the values of $a$ and $b$ for the exponential function $y=ab^{(x)}$ that passes through the points $(0,15)$ and $(4,1215)$ . Let's start by using the point $(0,15)$ . $\newline$ Substitute $x=0$ and $y=15$ into the equation $y=ab^{(x)}$ to find the value of $a$ . $\newline$ $b$ $0$ $\newline$ $b$ $1$ $\newline$ Since any number to the power of $b$ $2$ is $b$ $3$ , we have: $\newline$ $b$ $4$ $\newline$ Therefore, $b$ $5$ .
Find a Value of b: Now let's use the point $(4,1215)$ to find the value of b. $\newline$ Substitute $x=4$ , $y=1215$ , and $a=15$ into the equation $y=ab^{x}$ to find the value of b. $\newline$ $1215 = 15b^{4}$ $\newline$ To solve for b, divide both sides by $15$ : $\newline$ $\frac{1215}{15} = b^{4}$ $\newline$ $81 = b^{4}$ $\newline$ To find b, we take the fourth root of both sides: $\newline$ $b = 81^{\frac{1}{4}}$ $\newline$ $b = 3$
Write Exponential Function: Now that we have both $a$ and $b$ , we can write the exponential function. $a = 15$ and $b = 3$ , so the function is: $y = 15 \times 3^{x}$ This is the exponential function that goes through the points $(0,15)$ and $(4,1215)$ .

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