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Three points on the graph of the function
f(x) are
{(0,3),(1,9),(2,27)}. Which equation represents
f(x) ?

Three points on the graph of the function $f(x)$ are $\{(0,3),(1,9),(2,27)\}$ . Which equation represents $f(x)$ ?

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Write a polynomial from its roots

Full solution

Q. Three points on the graph of the function

f(x)

are

\{(0,3),(1,9),(2,27)\}

. Which equation represents

f(x)

?

Observe Pattern: Observe the given points to determine the pattern of the function. $\newline$ The points are $(0,3)$ , $(1,9)$ , and $(2,27)$ . We notice that as $x$ increases by $1$ , the $y$ -value seems to be multiplied by $3$ each time. This suggests an exponential pattern.
Test Exponential Function: Test the hypothesis that the function is exponential. If the function is exponential, it could be of the form $f(x) = a \cdot b^x$ . Since $f(0) = 3$ , we can determine the value of $a$ because any number to the power of $0$ is $1$ . So, $a \cdot 1 = 3$ , which means $a = 3$ .
Determine Value of a: Use another point to determine the value of b. Using the point $(1,9)$ , we substitute $x$ with $1$ and $f(x)$ with $9$ in the equation $f(x) = 3 \cdot b^x$ . We get $9 = 3 \cdot b^1$ , which simplifies to $b = \frac{9}{3} = 3$ .
Verify with Third Point: Verify the function with the third point. $\newline$ Using the point $(2,27)$ , we substitute $x$ with $2$ and $f(x)$ with $27$ in the equation $f(x) = 3 \times 3^x$ . We get $27 = 3 \times 3^2$ , which simplifies to $27 = 3 \times 9 = 27$ . This confirms our function is correct.

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