How do you do this:
Write an exponential function of the form y=ab^x that has a graph through (-2, 25) and (1, 10).
This is what I have so far:
10=ab^1
10/b=a
y=10/b*b^x
2/25=10b^-3
.008=b^-3
Then I am stuck. Am I doing it right so far? How do I continue? Thanks!
I don't get your second last line, I had:
25 = (10/b)b^-2
2.5 = b^-3
0.4 = b^3
b=cuberoot(.4)=.7368...
then a=10/.7368... = 13.572..
so y = 13.572(.7368)^x
I checked both data values, they work
gotcha thanks
You're close, but there seems to be a mistake in your calculations. Let's go through the steps together and find the correct answer.
Given the points (-2, 25) and (1, 10), we can substitute these values into the equation of an exponential function, y = ab^x, to find the values of a and b.
Step 1: Substituting (-2, 25) into the equation
25 = ab^(-2)
Step 2: Substituting (1, 10) into the equation
10 = ab^(1)
Now, let's solve these equations simultaneously to find the values of a and b.
Using the second equation, we can substitute "a" in terms of "b":
a = 10 / b.
Substituting this value of "a" into the first equation, we have:
25 = (10 / b) * b^(-2).
Simplifying the equation further, we get:
25 = 10 / b^2.
Now, we can multiply both sides of the equation by b^2 to isolate b :
25b^2 = 10.
Divide both sides by 25:
b^2 = 10/25.
Simplifying the right side gives:
b^2 = 2/5.
Taking the square root of both sides gives:
b = sqrt(2/5) ≈ 0.632.
Now that we have the value of b, we can substitute it back into either of the two original equations to find the value of a.
Using the equation 10 = ab^(1), we can substitute the values of a and b:
10 = a * 0.632.
Divide both sides by 0.632 to solve for a:
a = 10 / 0.632 ≈ 15.823.
Therefore, the exponential function that passes through the points (-2, 25) and (1, 10) is:
y = 15.823 * (0.632)^x.
I apologize for pointing out mistakes in your previous calculations. I hope this explanation clarifies the correct approach to finding the exponential function.