Eleventh Grade · Math · week 3

Transformations of functions

Apply shifts, reflections, and vertical stretches to a parent function, and predict a transformed function's behavior from its equation.

Every family of functions has a simple "parent" -- f(x)=x^2, f(x)=sqrt(x), f(x)=|x| -- and almost everything you graph this year is really just a parent function that has been moved, flipped, or stretched. Learning the transformation rules ONCE means you never have to re-derive a graph from scratch again; you just read the equation and picture the moves.

The rules: g(x) = f(x) + k shifts the graph up (k positive) or down (k negative). g(x) = f(x - h) shifts RIGHT when h is positive -- a common trap, since the sign looks backwards. g(x) = -f(x) reflects over the x-axis (flips it upside down). g(x) = a*f(x) stretches vertically when |a| > 1, and compresses when 0 < |a| < 1. Combine them and you can build almost any graph from its parent in a few mental steps.

📋 For the teacher
Teach it (10–15 min):
  • Build a "parent function toolkit" together: x^2 (parabola), sqrt(x) (half-parabola on its side), |x| (a V) -- sketch each one from memory before touching transformations.
  • Drill the backwards sign explicitly: have them evaluate g(5) for g(x)=f(x-5) and h(5) for h(x)=f(x+5), with f(x)=x^2, side by side, so they SEE which one matches shifting right.
  • For a stacked transformation like this week's example, peel it off in a fixed order every time: horizontal shift, reflection/stretch, vertical shift -- consistency beats speed at first.
  • Have them check one point by direct substitution (like g(0) in the example) every single time, as a built-in error check.
Talk about it:
  • Why does f(x-5) shift the graph RIGHT instead of left?
  • What is the difference between what "+k" outside the function does versus what it does inside?
  • If |a| is between 0 and 1 in g(x) = a*f(x), what happens to the graph?

Mastery looks like: They can describe every transformation in a combined equation in the correct order and evaluate the transformed function at a point.

Common stumbles: Reading f(x-h) as a shift in the same direction as the sign; applying a vertical stretch to the wrong part of a combined expression.

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