The general equation for an exponential function is y = a · bˣ where a and b are constants.
To compute a and b, use the information about which points the exponential graph passes through.
It passes through (3, 5) and (4, 10) That means when x = 3, y = 5 and when x = 4, y = 10
Plug these set of values into the generalized equation and solve for a and b
For (3, 5) we get 5 = ab³ (1) For (4, 10) we get 10 = ab⁴ (2)
Divide (2) by (1) ab⁴/ab³ = 10/5 b = 2
In equation (1) substituting b = 2 gives 5 = a · 2³ = 8a 8a = 5 a =5/8
Thus the exponential function is [tex]y = \dfrac{5}{8}\cdot2^x[/tex] (3)
Looking at the answer choices we see that there are only two distinct values of x that we need to compute value of y for. These are x =2 in choices A and B and x = 5 in choices C and D
Plug in x = 2 into equation (3): [tex]y = \dfrac{5}{8}\cdot2^2 \\\\y = \dfrac{5}{8}\cdot4\\\\y = 2.5[/tex]
None of these appear as a y-value in any of the choices. So eliminate A and B
Now find y corresponding to x = 5: [tex]y = \dfrac{5}{8}\cdot2^5 \\\\y = \dfrac{5}{8}\cdot32\\\\y = 5 \cdot 4\\\\y = 20[/tex]
This y value corresponds to the last choice for x = 5
