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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,16) and
(2,784).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,16)$ and $(2,784)$ . $\newline$ Answer:

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Write a linear equation from two points

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,16)

and

(2,784)

.

\newline

Answer:

Find 'a' value: Given points: $(0, 16)$ and $(2, 784)$ , we need to find the exponential function of the form $y = ab^x$ . We will use the first point to find the value of 'a' since any number raised to the power of $0$ is $1$ . Using the point $(0, 16)$ , we substitute $x = 0$ and $y = 16$ into the equation $y = ab^x$ to get: $16 = ab^0$ Since anything raised to the power of $0$ is $1$ , we have: $(2, 784)$ $2$ Therefore, $(2, 784)$ $3$ .
Find 'b' value: Now that we have the value of 'a', we can use the second point $(2, 784)$ to find the value of 'b'. We substitute $x = 2$ , $y = 784$ , and $a = 16$ into the equation $y = ab^x$ to get: $\newline$ $784 = 16b^2$ $\newline$ To solve for 'b', we divide both sides by $16$ : $\newline$ $\frac{784}{16} = b^2$ $\newline$ $49 = b^2$ $\newline$ Taking the square root of both sides gives us two possible solutions, $b = 7$ or $x = 2$ $0$ . Since we are dealing with an exponential function, which typically involves growth or decay, we consider only the positive value for 'b'. $\newline$ Therefore, $b = 7$ .
Write exponential function: We have found the values of 'a' and 'b': $a = 16$ and $b = 7$ . We can now write the equation of the exponential function: $\newline$ $y = 16 \times 7^x$ $\newline$ This is the exponential function that passes through the points $(0, 16)$ and $(2, 784)$ .

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