Answer :
Answer:
[tex]y=21\left(\dfrac{1}{7}\right)^x[/tex]
Step-by-step explanation:
Exponential Equations
Exponential functions or equations are expressions that show either exponential growth or decay and have the format of,
[tex]y=a(b)^x[/tex],
where a is the y-intercept, b is the growth/decay factor (if it's growth then, b > 1; if it's decay then, 0 < b < 1).
[tex]\hrulefill[/tex]
Solving the Problem
Algebraic Solution:
We're given two solutions to our mystery exponential function: (0, 21) and (1, 3). We can plug them into the general formula and solve for a and b!
So,
[tex]21=a(b)^0[/tex]
and
[tex]3=a(b)^1[/tex].
Solving the first equation we get,
[tex]21=a(1)\\\Longrightarrow a=21[/tex].
We can plug the a value into the second equation to get b,
[tex]3=21(b)^1\\\\3=21b\\\Longrightarrow \dfrac{1}{7}=b[/tex].
So our function is,
[tex]y=21\left(\dfrac{1}{7}\right)^x[/tex].
Geometric Solution:
Knowing that all y-intercepts have an x-value of 0, we can identify that one of the points or solutions given in the problem is our y-intercept: (0, 21). Thus, a = 21.
From the two points given, we can see that as x gets bigger in the positive direction, the y-value decreases indicating that this is a decay function.
To be specific, we see that the y-value decreases from 21 to one-seventh of its value: 3. So, b = 1/7.
Plugging in our a and b values into the function we get,
[tex]y=21\left(\dfrac{1}{7}\right)^x[/tex].
