StellarIn Unit 1.2 of AP Precalculus, you look closer at how specific types of functions change. For a linear function, the average rate of change is constant over any interval. This means that for equal increases in the input variable x, the output variable y changes by the exact same amount every single time. This constant rate of change is the slope of the line, represented by:
For a quadratic function, the average rate of change is not constant—it changes linearly. If you look at consecutive intervals of equal width, the rate of change increases or decreases by a constant amount
When you analyze how these rates change over consecutive intervals for a quadratic function, it tells you about the function's opening direction:
- Opening upward (concave up): The average rate of change is increasing over consecutive intervals.
- Opening downward (concave down): The average rate of change is decreasing over consecutive intervals.
This gives you a way to mathematically prove whether a function is linear or quadratic just by looking at a table of values and calculating the rates of change across constant intervals.