For the exponential function represented in the table below

For the exponential function represented in the table below, determine the initial value, the n -unit growth factors, and the exponential function that models the data. The initial value: The 1-unit growth factor: The 1-unit percent change: The 3-unit growth factor: The l/4-unit growth factor: Function definition:

Solution

We define an exponential function to be any function of the form:

y = y₀ · m ^x, with initial value = y_0.

(a) Initial Value i.e. at x = 0, y = y₀, at x = a, y₁= m y₀, y₂ = m² y₀, y₃ = m³ y₀

Now, [y₄/y₁]=634.35/788.64 = 0.804 = m³, Therefore, m = 0.93

y₀ = y₁/m = 788.64/0.93 = 848

(b) One unit growth factor = m = 0.93

(c) One unit % change = 0.93 (788.64 - 848) x 100 / 848 = -6.51%

(d) Three unit growth factor

y = y₀ m^x+3 = (y₀ · m ^x) m³, therefore m³ = (0.93)³ = 0.804

(e) The 1/4 growth factor = m^1/4 = 0.93^1/4 =  0.982

(f) Fuction y = 848 (0.93)^x, x>=0.