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Write an exponential function in the form
y=ab^(x) that goes through the points
(0,16) and
(9,8192).
Answer:

Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,16)$ and $(9,8192)$ . $\newline$ Answer:

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Write and solve equations that represent diagrams

Full solution

Q. Write an exponential function in the form

y=a b^{x}

that goes through the points

(0,16)

and

(9,8192)

.

\newline

Answer:

Find 'a' value: Use the first point $(0,16)$ to find the value of 'a'. $\newline$ The general form of an exponential function is $y = ab^x$ . If we plug in $x = 0$ and $y = 16$ , we get: $\newline$ $16 = ab^0$ $\newline$ Since any number to the power of $0$ is $1$ , we have: $\newline$ $16 = a \times 1$ $\newline$ Therefore, $a = 16$ .
Find 'b' value: Use the second point $(9,8192)$ to find the value of 'b'. $\newline$ Now we know that $a = 16$ , we can substitute this value into the equation using the second point $(9,8192)$ : $\newline$ $8192 = 16b^9$ $\newline$ To solve for $b$ , we divide both sides by $16$ : $\newline$ $\frac{8192}{16} = b^9$ $\newline$ $512 = b^9$ $\newline$ Now we need to find the ninth root of $512$ to solve for $b$ : $\newline$ $a = 16$ $0$ $\newline$ Calculating the ninth root of $512$ gives us: $\newline$ $a = 16$ $2$
Write final function: Write the final exponential function. $\newline$ Now that we have both $a$ and $b$ , we can write the exponential function: $\newline$ $y = 16 \times 2^x$ $\newline$ This is the function that goes through the points $(0,16)$ and $(9,8192)$ .

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