Mathematics.

functions

Function Transformations

Algebra I30 minDifficulty3 out of 10

You should know: functions

Overview

Function transformations describe how the graph of a function changes when you modify its equation — shifting it up, down, left, or right, stretching or compressing it, or reflecting it across an axis. Learning these transformation rules lets you sketch complicated-looking functions instantly by starting from a simple parent function (like y=x² or y=√x) and applying a short sequence of predictable moves.

Intuition

Every transformation follows one consistent, slightly counterintuitive rule: changes made OUTSIDE the function (added to or multiplied onto the whole expression) move the graph the way you'd expect — add moves it up, multiply stretches it vertically. But changes made INSIDE the function (applied directly to x before the function acts on it) move the graph the OPPOSITE of what you'd guess — replacing x with x-3 shifts the graph RIGHT by 3, not left, because the function now needs a bigger input to produce the same output it used to produce at x.

Interactive Graph

Drag h, k to shift the parabola

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Formal Definition

Definition

Given a base function f(x), the transformed function g(x) = a·f(b(x-h)) + k combines all four transformation types:

g(x)=a⋅f(x−h)+kg(x) = a\cdot f(x-h) + k

h shifts horizontally (right if h>0), k shifts vertically (up if k>0)

Shift
g(x)=a⋅f(x),g(x)=f(bx)g(x) = a\cdot f(x), \quad g(x)=f(bx)

|a|>1 stretches vertically, 0<|a|<1 compresses; a<0 reflects over the x-axis; b<0 reflects over the y-axis

Stretch/compress and reflect

Properties

Vertical shift

f(x)+k: shifts up k units if k>0f(x)+k: \text{ shifts up } k \text{ units if } k>0

Horizontal shift

f(x−h): shifts right h units if h>0f(x-h): \text{ shifts right } h \text{ units if } h>0

Vertical stretch/compression

a⋅f(x): stretches away from x-axis if ∣a∣>1, compresses toward it if 0<∣a∣<1a\cdot f(x): \text{ stretches away from x-axis if } |a|>1, \text{ compresses toward it if } 0<|a|<1

Reflections

−f(x) reflects over the x-axis;f(−x) reflects over the y-axis-f(x) \text{ reflects over the x-axis}; \quad f(-x) \text{ reflects over the y-axis}

Applications

Shifting a wave function horizontally models a phase delay; stretching it vertically models a change in amplitude.

Worked Examples

  1. The -5 is added OUTSIDE the squaring, so it's a vertical shift.

    g(x)=f(x)−5g(x) = f(x) - 5

Answer: Vertical shift down 5 units

Practice Problems

Difficulty 3/10

Describe the transformation from f(x) = |x| to g(x) = |x - 4| + 2.

Difficulty 4/10

Which transformation describes g(x) = f(2x)?

Common Mistakes

Common Mistake

Assuming f(x+3) shifts the graph right, by analogy with how +3 outside the function shifts up.

Transformations INSIDE the function work opposite to intuition: f(x+3) shifts LEFT by 3, and f(x-3) shifts RIGHT by 3. Only transformations OUTSIDE the function (added or multiplied to the whole f(x)) behave in the 'expected' direction.

Common Mistake

Confusing a vertical stretch with a horizontal compression, since both make a graph look 'taller/narrower.'

A vertical stretch (a·f(x), |a|>1) multiplies OUTPUTS, making the graph taller at every x. A horizontal compression (f(bx), |b|>1) multiplies INPUTS, squeezing the graph toward the y-axis — they affect different axes and are computed differently.

Summary

  • Transformations outside f(x) behave intuitively: +k shifts up, ×a stretches vertically.
  • Transformations inside f(x) behave opposite to intuition: f(x-h) shifts RIGHT by h.
  • Negative a reflects over the x-axis; negative b (inside) reflects over the y-axis.
  • g(x) = a·f(b(x-h)) + k combines all four transformation types in one expression.

References

  1. WebsiteWikipedia — Function (mathematics), Transformations