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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,19) and
(7,2432).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,19)$ and $(7,2432)$ . $\newline$ Answer:

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Write an equivalent ratio

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,19)

and

(7,2432)

.

\newline

Answer:

Find 'a' value: To find the exponential function, we need to determine the values of 'a' and 'b' in the equation $y=ab^{x}$ . We can use the point $(0,19)$ to find 'a' because any number raised to the power of $0$ is $1$ . So, the equation becomes $y=a\cdot 1$ , which simplifies to $y=a$ . Since the function passes through $(0,19)$ , we can set $y$ to $19$ and solve for 'a'. $\newline$ Calculation: $19 = a \cdot 1$ $\newline$ $(0,19)$ $0$
Find 'b' value: Now that we have the value of 'a', we need to find 'b'. We can use the second point $(7,2432)$ to find 'b'. We substitute 'a' with $19$ in the equation $y=ab^{x}$ and then solve for 'b'. $\newline$ Calculation: $2432 = 19 \cdot b^{7}$ $\newline$ $b^{7} = \frac{2432}{19}$ $\newline$ $b^{7} = 128$
Calculate 'b': To find 'b', we need to take the seventh root of $128$ . The seventh root of a number is the same as raising that number to the power of $\frac{1}{7}$ . $\newline$ Calculation: $b = (128)^{\frac{1}{7}}$ $\newline$ $b \approx 2$
Check approximation: We need to check if $b \approx 2$ is a correct approximation. Since $2^7 = 128$ , we can confirm that $b = 2$ is the exact value and not just an approximation. $\newline$ Calculation: $2^7 = 128$
Write exponential function: Now that we have both $a$ and $b$ , we can write the exponential function. We have $a = 19$ and $b = 2$ , so the function is $y = 19 \times 2^{x}$ .

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