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 [tex]y=a(b) ^{x} [/tex]. 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. [tex]14=a(b) ^{4} [/tex] and [tex]16=a(b) ^{5} [/tex]. 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: [tex] \frac{14}{b ^{4} } =a[/tex] and [tex] \frac{16}{b ^{5} } =a[/tex]. 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: [tex] \frac{14}{b ^{4} } = \frac{16}{b ^{5} } [/tex]. Cross multiply to get [tex]14b ^{5} =16b ^{4} [/tex]. Now divide each side by the b with the lower exponent so we can cancel one of them out. [tex] \frac{14b ^{5} }{b ^{4} } = \frac{16b ^{4} }{b ^{4} } [/tex]. The b^4 cancels by the 16 and reduces by the 14 to leave you with this: [tex]14b=16[/tex], and b = 8/7, or 1.14 rounded. Now we will sub that in to one of our equations and solve for a. [tex]14=a( \frac{8}{7}) ^{4} [/tex] and [tex]14=a( \frac{4096}{2401} )[/tex]Multiply both sides by 2401 to find that a = 8.21 rounded. So your exponential equation is [tex]y=8.21(1.14) ^{x} [/tex]. a = 8.21 is the growth factor.
