Write the exponential equation that contains the points (−2,4)

Question

(
−
2
,
4
)
and (3,972)
(
3
,
972
)
.(1 point)
The equation is $$y=·.

Answer 1

To find the exponential equation that passes through the points $(-2,4)$ and $(3,972)$, we can assume the general form of the exponential equation is:

$$y=a{b}^x$$

where $a$ is the initial value (the value of $y$ when $x=0$) and $b$ is the growth factor.

We can create two equations using the given points:

1. For the point $(-2,4)$: $$4=a{b}^{-2}$$

2. For the point $(3,972)$: $$972=a{b}^3$$

Now we have a system of two equations:

1. $4=\frac{a}{b^2}$
2. $972=a{b}^3$

From the first equation, we can express $a$ in terms of $b$: $$a=4b^2$$

Now, substitute this expression for $a$ into the second equation: $$972=(4b^2)b^3$$ $$972=4b^5$$

Now, solve for $b$: $$b^5=\frac{972}{4}=243$$

To solve for $b$, take the fifth root of $243$: $$b={243}^{1/5}=3$$

Now that we have $b$, we can find $a$ using $a=4b^2$: $$a=4(3^2)=4\times 9=36$$

Now we have both $a$ and $b$: $$a=36,b=3$$

So, the exponential equation that contains the points $(-2,4)$ and $(3,972)$ is: $$y=36\cdot 3^x$$

The final equation is: $$\boxed{{\displaystyle y=36\cdot 3^x}}$$