Question

Suppose that f is an exponential function with f (4) = 14 and f (5) = 16. What is the growth factor for f?

Answer 1

You have to do this in quite a few steps.  Your two points, written in coordinate form, are (4, 14) and (5, 16).  The standard form for an exponential function is $y=a(b) ^{x}$.  Since neither one of these gives us an initial value of (0, a), we have to sub in the two points into 2 equations, creating a system that we can then solve for a, then b.  Our system will look like this after we sub in the x and y values from each point. $14=a(b) ^{4}$ and $16=a(b) ^{5}$.  We need to solve for a and b.  Let's solve for a first.  Do that in each equation by dividing by the b^power: $\frac{14}{b ^{4} } =a$ and $\frac{16}{b ^{5} } =a$.  By the symmetric property, if equation 1 = a and equation 2 = a, then equation 1 = equation 2.  So let's set them equal to each other: $\frac{14}{b ^{4} } = \frac{16}{b ^{5} }$.  Cross multiply to get $14b ^{5} =16b ^{4}$.  Now divide each side by the b with the lower exponent so we can cancel one of them out. $\frac{14b ^{5} }{b ^{4} } = \frac{16b ^{4} }{b ^{4} }$.  The b^4 cancels by the 16 and reduces by the 14 to leave you with this: $14b=16$, and b = 8/7, or 1.14 rounded.  Now we will sub that in to one of our equations and solve for a. $14=a( \frac{8}{7}) ^{4}$ and $14=a( \frac{4096}{2401} )$Multiply both sides by 2401 to find that a = 8.21 rounded.  So your exponential equation is $y=8.21(1.14) ^{x}$.  a = 8.21 is the growth factor.