The table below shows some values of the exponential function f, where f(t) = p \cdot q^t … — SAT

The table below shows some values of the exponential function $f$ , where $f(t) = p \cdot q^t$ for constants $p$ and $q$ .

What is the value of $q$ ?

Answer: 1.5

1.5

The correct answer is 1.5. Since the value of the function at

t = 0

80

, the initial value coefficient is

80

. At

t = 2

, the value is

180

, which gives the equation

80 \cdot q^2 = 180

. Solving for

q^2

yields

q^2 = 2.25

, and taking the positive square root gives

q = 1.5

Set up the general exponential equation using the initial value

f(0) = p \cdot q^0 = 80 \implies p = 80

The initial value at

t = 0

directly gives the coefficient

p

because

q^0 = 1

Substitute another point from the table to solve for the base

q

f(2) = 80 \cdot q^2 = 180 \implies q^2 = 2.25

Using the point

(2, 180)

allows us to write an equation with one variable,

q

Solve for

q

by taking the square root

q = 1.5

Since the base of an exponential function must be positive, we take the positive square root of 2.25.

Determining the base of an exponential function from a table of values

Question