Four Representations — The AP Framework

The AP exam emphasizes understanding functions in four ways:

1. Graphical: What does the graph tell you?

2. Numerical: What do the table values tell you?

3. Analytical: What does the equation tell you?

4. Verbal: What does the word problem describe?

You must translate between all four.

Given a graph → find the derivative at a point (slope of tangent)

Given a table → approximate an integral (Riemann sum)

Given an equation → find extrema (set derivative = 0)

Given words → set up the equation or integral

The hardest AP questions combine multiple representations. "Here's a graph of f'. Here's a table of g values. Find d/dx[f(g(x))] at x = 2." You need chain rule + graph reading + table lookup.

Reference:

Wikipedia: AP Calculus

https://en.wikipedia.org/wiki/AP_Calculus