Function Transformation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Function Transformation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function transformation shifts, stretches, compresses, or reflects the graph of a parent function by modifying its formula in a systematic way.

Moving or reshaping a graph without changing its basic shape.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Changes inside f(x-h) affect x (horizontal). Changes outside f(x)+k affect y (vertical).

Common stuck point: f(x - 3) shifts RIGHT (opposite of the sign). f(x) - 3 shifts DOWN.

Sense of Study hint: Compare f(x) and the transformed version by plugging in the same x-values. Notice which direction the graph moved.

Worked Examples

Example 1

easy
Describe all transformations applied to f(x) = x^2 to obtain g(x) = 2(x-3)^2 + 1.

Solution

  1. 1
    Write in standard form y = a\cdot f(b(x-h))+k: here a=2, b=1, h=3, k=1.
  2. 2
    Horizontal shift: h=3 shifts the parabola 3 units to the right (vertex moves from (0,0) to (3,0)).
  3. 3
    Vertical stretch: a=2 stretches vertically by factor 2 (makes parabola narrower). Vertical shift: k=1 shifts the entire graph 1 unit up. Final vertex: (3, 1).

Answer

Shift right 3, stretch vertically by 2, shift up 1; vertex at (3,1)
The transformation y = a\cdot f(b(x-h))+k encodes four independent transformations. Reading off a, b, h, k allows systematic description without re-deriving the graph from scratch.

Example 2

medium
Starting from f(x) = \sqrt{x}, apply the transformation g(x) = -\sqrt{2x+4} step by step and identify the key point transformations.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The graph of y = f(x) passes through (1, 4). Where does the transformed graph y = f(x+2) - 3 pass through?

Example 2

hard
Write the equation of the function obtained by reflecting f(x)=\log_2(x) over the y-axis, compressing horizontally by a factor of 3, and shifting up 5.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitioncoordinate plane