Unit 22.2 Change in Linear and Exponential FunctionsNo calculator

The plotted points represent outputs at equal input intervals. Which conclusion is supported?

Discrete points at (0,2), (1,3), (2,4.5), and (3,6.75), joined only to guide the eye.

Linear, because each output is larger
Linear, because the changes are $1,1.5,2.25$
Exponential, because the first differences increase
Exponential, because each output is $1.5$ times the previous output

Answer and derivation

Correct answer: D

A constant output ratio of $1.5$ supports an exponential function.

The first differences are not constant.
Each successive output is multiplied by $1.5$ .
Constant proportional change over equal input intervals is exponential.

Why the other options fail

A: Increasing output does not distinguish model type.
B: Those unequal differences contradict linearity.
C: Increasing differences alone could also occur for other nonlinear models.
D: Correct constant-ratio evidence.

Common misconception: Curvature or increasing differences alone do not establish an exponential model.

China/AP bridge: China-course bridge: linear and exponential change build on arithmetic and geometric patterns; AP requires evidence across representations; ratio evidence is stronger than visual shape; avoid calling every rising curve exponential.