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Find the equation of the exponential given
(2,2) and
(3,4)

$2$ . Find the equation of the exponential given $(2,2)$ and $(3,4)$

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Write a polynomial from its roots

Full solution

Q.

2

. Find the equation of the exponential given

(2,2)

and

(3,4)

Identify Exponential Function Form: Identify the general form of an exponential function, which is $y = a b^{x}$ . Here, $a$ and $b$ are constants we need to find.
Substitute Point $2,2$ : Use the point $2,2$ in the equation $y = ab^x$ . Substituting $x = 2$ and $y = 2$ gives $2 = ab^2$ .
Substitute Point $(3,4)$ : Use the point $(3,4)$ in the equation $y = ab^x$ . Substituting $x = 3$ and $y = 4$ gives $4 = ab^3$ .
Solve System of Equations: Solve the system of equations from the two points. From the first equation, express $a$ as $a = \frac{2}{b^2}$ . Substitute this into the second equation: $4 = \left(\frac{2}{b^2}\right)b^3$ . Simplify to get $4 = 2b$ .
Find Value of b: Solve for $b$ : $2b = 4$ , so $b = 2$ .
Find Value of a: Substitute $b = 2$ back into the expression for a: $a = \frac{2}{2^2} = \frac{2}{4} = 0.5$ .
Write Final Equation: Write the final equation using the values of $a$ and $b$ : $y = 0.5 \times 2^x$ .

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