Comparing Quadratic Functions Across Representations

In this lesson you will: Compare two quadratic functions given as equations, graphs, tables or descriptions; Extract the same feature from each representation before comparing anything; Decide which function is greater at a given input, and say where that changes.

Put both functions into the same language first

You cannot compare a graph with a table by looking back and forth between them. Decide which feature the question is about — vertex, maximum, minimum, intercepts, direction — then pull that one feature out of each representation and compare two numbers.

The work is almost never hard. The mistakes come from comparing a feature of one function with a different feature of the other.

A table gives itself away in its second differences

When the inputs of a table are equally spaced, subtract consecutive outputs to get first differences, then subtract those to get second differences. A quadratic has constant second differences, and that constant equals 2a.

In the table below the second differences are all -2, so 2a=-2 and a=-1. The outputs turn around at x=3, so the vertex is (3,4) — the function is f(x)=-x^2+6x-5.

Which one is greater depends on where you ask

\"Which function is bigger\" is not a question a parabola can answer on its own. Two quadratics can trade places more than once, so the honest answer names an input.

Take f(x)=x^2-6x+11, given by an equation, and g, shown in the figure. Completing the square, f(x)=(x-3)^2+2, so f has a minimum value of 2 at x=3. The graph of g has vertex (1,-4). So g has the lower minimum — but at x=6, f(6)=11 while g(6)=(6-1)^2-4=21, and g is far above.

Common mistake — Comparing whichever number is easiest to see

f is given by a graph with vertex (2,-3). g is given by the equation g(x)=x^2-6x+5. Asked which has the lower minimum, a student compares -3 with 5 and answers f.

The 5 was simply the number that was easy to read. It is g's y-intercept, not its minimum.

g's axis is x=3 and g(3)=9-18+5=-4, so g's minimum is -4 — lower than f's -3. The correct answer is the opposite of the one the shortcut produced.

How to catch it: Say the comparison out loud with the feature named twice: \"minimum value of f against minimum value of g.\" If you cannot say where each number came from, one of them has not been computed yet.