Question

quadratic data
ticket price (dollar x) | profit (billions of dollars)
200 | 3.08
250 | 3.52
300 | 3.76
350 | 3.82
400 | 3.70
450 | 3.38
analyze the data represented in the table and select the appropriate model.
a exponential
b linear
c none
d quadratic

Explanation:

Step1: Check Linear Trend

Calculate the differences in profit (Δy) for each Δx (50, since ticket price increases by 50 each time).
For x=200 to 250: Δy = 3.52 - 3.08 = 0.44
x=250 to 300: Δy = 3.76 - 3.52 = 0.24
x=300 to 350: Δy = 3.82 - 3.76 = 0.06
x=350 to 400: Δy = 3.70 - 3.82 = -0.12
x=400 to 450: Δy = 3.38 - 3.70 = -0.32
Differences are not constant, so not linear.

Step2: Check Quadratic Trend (Second Differences)

Calculate second differences (Δ²y) from the first differences:
First differences: 0.44, 0.24, 0.06, -0.12, -0.32
Second differences: 0.24 - 0.44 = -0.20; 0.06 - 0.24 = -0.18; -0.12 - 0.06 = -0.18; -0.32 - (-0.12) = -0.20
Second differences are approximately constant (close to -0.20), indicating a quadratic model (since quadratic functions have constant second differences).

Step3: Eliminate Exponential

Exponential models have multiplicative growth/decay, but here the profit first increases then decreases, not exponential growth/decay.

Answer:

D. quadratic