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There is an exponential function that is $y$ positive and decreasing. Two coordinates are $(0, 3600)$ and $(2,225)$ . What is the equation

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Find derivatives of exponential functions

Full solution

Q. There is an exponential function that is

y

positive and decreasing. Two coordinates are

(0, 3600)

and

(2,225)

. What is the equation

Identify Form of Function: Identify the general form of the exponential function: $y = a b^{x}$ . Here, $a$ is the initial value, and $b$ is the base of the exponential function.
Find Initial Value 'a': Use the given point $(0, 3600)$ to find 'a'. Substituting $x = 0$ and $y = 3600$ into the equation $y = ab^x$ gives $3600 = ab^0$ . Since anything raised to the power of $0$ is $1$ , we have $3600 = a \times 1$ , so $a = 3600$ .
Find Base 'b': Use the second point $(2, 225)$ and the value of 'a' to find 'b'. Substituting $x = 2$ and $y = 225$ into the equation $y = 3600b^2$ gives $225 = 3600b^2$ . Solving for $b$ , we get $b^2 = \frac{225}{3600} = \frac{1}{16}$ . Taking the square root of both sides, $b = \frac{1}{4}$ or $b = -\frac{1}{4}$ . Since the function is decreasing, we choose $b = \frac{1}{4}$ .
Write Final Equation: Write the final equation using the values of $a$ and $b$ . The equation of the exponential function is $y = 3600(\frac{1}{4})^x$ .

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