Transformations of Functions

Every change to a function's equation moves its graph in a predictable way.

Outside the function (affecting outputs) behaves as you expect:

• f(x) + k shifts up k
• a·f(x) stretches vertically by a; a negative a flips it upside down

Inside the function (affecting inputs) behaves backwards:

• f(x − h) shifts *right* h, not left
• f(bx) squeezes horizontally by 1/b
• f(−x) flips it left-right

Why backwards? Because f(x − 3) has to wait until x reaches 3 to do what f did at 0. The graph is delayed, so it moves right.

Order matters: stretch and reflect before shifting.

Write a transformed function as g(x) = a·f(b(x − h)) + k.

Vertical transformations act on the output and compose in the natural order: multiply by a, then add k. Horizontal transformations act on the input and are therefore *inverted*, both in effect and in order. To reach a given output, the input must be pre-processed, so a subtraction inside produces a rightward shift and a coefficient b produces a compression by 1/b.

Formally, (x, y) on f maps to (x/b + h, ay + k) on g, which is the reliable way to transform specific points such as a vertex or an endpoint.

Sign placement matters: −f(x) reflects across the x-axis while f(−x) reflects across the y-axis. Care is needed with an expression like f(2x − 6), which must be factored to f(2(x − 3)) before the shift can be read as 3 rather than 6.

These rules apply to every family identically, which is what makes the parent-function approach efficient.