Help needed!
Find the base of the exponential function whose graph contains the given points.
1. (2,16)
2. (1,10)
3. (3,64)
4. (3,1/343)
5. (4,1/625)
6. (3/2,27)
2^4 = 16
10^1 = 10
4^3 = 64
and so on.
(3,64)
What is the base of the exponential function whose graph contains the point (2,16)
1. I'm on it! Let me put on my math clown shoes here. *honk honk* So, to find the base of an exponential function, we need to remember that our equation looks like this: y = ab^x. Now, let's plug in our first point (2,16). We have 16 = ab^2. Are you with me so far?
2. Alright, next up is the point (1,10). Let's substitute these values into our equation. We get 10 = ab^1.
3. Time for the third point, (3,64). Substitute these values again. We have 64 = ab^3.
4. Moving on to the point (3,1/343). Let's plug it in. We get 1/343 = ab^3.
5. For the fifth point, (4,1/625), we have 1/625 = ab^4.
6. Finally, for the last point (3/2,27), we have 27 = ab^(3/2).
Now, we have a system of equations! Let's see if we can clown our way through solving it. *squeaky honk*
To find the base of an exponential function, we can use the general form of an exponential function:
f(x) = a * b^x
where a is the initial value and b is the base.
Let's go through each of the given points and find the base of the exponential function for each case:
1. (2,16):
Plug in the values into the function: 16 = a * b^2
We don't have the value of 'a,' but we can find the ratio of neighboring y-values:
16/10 = (a * b^2) / (a * b^1)
1.6 = b
Therefore, the base of the exponential function is 1.6.
2. (1,10):
Following the same logic, we have: 10/16 = (a * b^1) / (a * b^2)
0.625 = b
Therefore, the base of the exponential function is 0.625.
3. (3,64):
64/16 = (a * b^3) / (a * b^2)
4 = b
Therefore, the base of the exponential function is 4.
4. (3,1/343):
(1/343) / 16 = (a * b^3) / (a * b^2)
(1/20,608) = b
Therefore, the base of the exponential function is 1/20,608.
5. (4,1/625):
(1/625) / (1/343) = (a * b^4) / (a * b^3)
(1/49) = b
Therefore, the base of the exponential function is 1/49.
6. (3/2,27):
27/16 = (a * b^(3/2)) / (a * b^2)
(27/16) = b^(1/2)
Taking the square root of both sides, we get: sqrt(27/16) = b^(1/2)
(sqrt(27)/sqrt(16)) = b^(1/2)
(3sqrt(3)/4) = b
Therefore, the base of the exponential function is 3sqrt(3)/4.
So, the bases for the given points are:
1. 1.6
2. 0.625
3. 4
4. 1/20,608
5. 1/49
6. 3sqrt(3)/4