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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,19) and
(3,6517).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,19)$ and $(3,6517)$ . $\newline$ Answer:

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Solve multi-variable equations

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,19)

and

(3,6517)

.

\newline

Answer:

Find 'a' value: Use the point $(0,19)$ to find the value of 'a'. $\newline$ Since the point $(0,19)$ lies on the graph of the function, we can substitute $x=0$ and $y=19$ into the equation $y=ab^{x}$ to find 'a'. $\newline$ $y = ab^{0}$ $\newline$ $19 = a \cdot b^0$ $\newline$ Since any number raised to the power of $0$ is $1$ , we have: $\newline$ $19 = a \cdot 1$ $\newline$ Therefore, $(0,19)$ $0$ .
Find 'b' value: Use the point $(3,6517)$ to find the value of 'b'. $\newline$ Now that we know 'a', we can substitute $x=3$ , $y=6517$ , and $a=19$ into the equation to solve for 'b'. $\newline$ $6517 = 19 \times b^{3}$ $\newline$ To isolate $b$ , we divide both sides by $19$ : $\newline$ $\frac{6517}{19} = b^{3}$ $\newline$ $343 = b^{3}$ $\newline$ To find $b$ , we take the cube root of both sides: $\newline$ $x=3$ $0$ $\newline$ $x=3$ $1$
Write final exponential function: Write the final exponential function. $\newline$ Now that we have both $a$ and $b$ , we can write the exponential function: $\newline$ $y = ab^{x}$ $\newline$ $y = 19 \times 7^{x}$ $\newline$ This is the exponential function that goes through the points $(0,19)$ and $(3,6517)$ .

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