Math 1 · Unit 7 · Lesson 5
Linear or Exponential?
From a table: check that x goes up by equal steps, subtract to look for a constant difference (linear), and if that fails divide to look for a constant ratio (exponential). From a story: a fixed amount per step adds, while doubles, halves, and percents multiply. From a graph: a line stays a line, and an exponential growth curve eventually passes any line, no matter how steep.
The big idea
Lines add and exponentials multiply, so the test is subtract, then divide. First make sure x goes up by equal steps. Subtract each output from the next: a constant difference means linear, and the difference is the slope. If the differences change, divide each output by the one before: a constant ratio means exponential, and the ratio is b. If neither is constant, it is neither family. Stories give the same answer through their words: a fixed amount per step adds, while doubles, halves, and percents multiply, so grows by 10 is linear and grows by 10% is exponential. Because a line adds the same amount forever while the exponential's step keeps growing, exponential growth eventually exceeds any line, which is why the doubling penny beats a dollar a day.
The parts of this lesson
Each part is its own short page. Go in order, or pick the one you need.
Printables
One page with the four-step table test, the clue words, the Town A versus Town B race, and the doubling penny. Plus a worksheet with an answer key and a short lesson plan.