Find the exponential function f(x)=Ca^x whose graph goes through the points (0,3) and (2,12).
for (03) --> 3 = C a^0
3 = C (1)
so C = 3
for (2,12) --- 12 = 3a^2
4 = a^2
a = 2
f(x) = 3 (2)^x
Well, let me tell you, finding exponential functions can be exponential-lly fun! So, let's get to it, shall we?
Given the points (0,3) and (2,12), we can set up a system of equations to solve for C and a.
First, let's plug in the coordinates of the point (0,3):
3 = Ca^0
3 = C * 1
3 = C
So, we have found the value of C to be... drum roll please... 3!
Now, let's plug in the coordinates of the point (2,12) into our function with C = 3:
12 = 3 * a^2
4 = a^2
a = √4
a = 2 or a = -2
Since exponential functions can't have negative bases (that would make it super imaginary, like a unicorn fountain that shoots out rainbow lasers), we'll go with a = 2.
So, the final form of the exponential function f(x) = 3 * 2^x will make those two points happy campers.
Keep in mind, this function will only be true for the given points. If any more points come into the picture, it's back to the math lab for us!
And there you have it, my friend! An exponential function that's bound to put a smile on your face.
To find the exponential function that passes through the points (0,3) and (2,12), we can follow these steps:
Step 1: Use the point (0,3) to find the value of "C".
Since f(0) = 3, we substitute x = 0 into the function:
3 = Ca^0
3 = C * 1
C = 3
Step 2: Use the point (2,12) to find the value of "a".
Since f(2) = 12, we substitute x = 2 into the function:
12 = 3a^2
Divide both sides of the equation by 3:
4 = a^2
Taking the square root of both sides, we get:
a = ±2
However, since exponential functions cannot have negative bases, we choose a = 2.
Therefore, the exponential function f(x) = 3 * 2^x passes through the points (0,3) and (2,12).
To find the exponential function that goes through the points (0,3) and (2,12), we first need to determine the values of C and a.
The general form of an exponential function is f(x) = Ca^x, where C is a constant and a is the base.
Using the first point (0,3), we substitute the values of x and f(x) into the equation:
3 = Ca^0
3 = C * 1 (since a^0 is always 1)
C = 3
Now, we have the value of C.
Using the second point (2,12), we substitute the values of x and f(x) into the equation:
12 = 3a^2 (since C is 3, substituting into the general form)
4 = a^2 (dividing both sides by 3)
Now, we have the value of a.
To find the exact value of a, we can take the square root of both sides of the equation:
√4 = √(a^2)
2 = a
So, the exponential function f(x) = Ca^x that passes through the points (0,3) and (2,12) is:
f(x) = 3 * 2^x