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Graph transformations
Transformations of functions
CatalogueDPMathematicsMathematics: Analysis and Approaches HLFunctions

Transformations of functions

Predict how changes inside and outside a function move, stretch and reflect its graph.

  • AA HL
  • Graph transformations
  • Papers 1 and 2
  • GDC useful for checking
  • About 20 min

By the end of this lesson

  • distinguish horizontal from vertical transformations
  • apply shifts, stretches and reflections
  • track the order of combined transformations

Why this matters. Graph transformations let you reason from a known function instead of rebuilding every graph point by point.

Inside changes inputs; outside changes outputs

Outside $f$

$f(x)+a$: up $a$

$af(x)$: vertical scale factor $|a|$

Inside $f$

$f(x-a)$: right $a$

$f(ax)$: horizontal scale factor $1/|a|$

Graph transformationTwo parabolas show y equals x squared and y equals the quantity x minus two squared plus one, shifted two units right and one unit up.y = x²y = (x−2)² + 1
The vertex moves from $(0,0)$ to $(2,1)$: right two, then up one.

Checkpoint

Starting from $y=f(x)$, what does $y=f(x+3)$ do?
Show explanation

To obtain the old input $0$, the new coordinate must be $x=-3$, so points move left.

Example 1

Track a key point

From $f(x)$ to $-2f(x-4)+1$

If $(1,3)$ lies on $y=f(x)$, then the horizontal shift sends $x=1$ to $x=5$. The output changes from $3$ to $-2(3)+1=-5$.

$$(1,3)\longmapsto(5,-5).$$

This point mapping is safer than relying on a memorized sketch.

Combine transformations carefully

For $y=a f(b(x-h))+k$, horizontal coordinates change by $x\mapsto x/b+h$, while outputs change by $y\mapsto ay+k$.

Order the method

For $y=3f(2(x-1))-4$, order the operations applied to points of $y=f(x)$.

  1. Compress horizontally by factor $1/2$
  2. Shift right $1$
  3. Stretch vertically by factor $3$
  4. Shift down $4$
Show a valid order
  1. Compress horizontally by factor $1/2$
  2. Shift right $1$
  3. Stretch vertically by factor $3$
  4. Shift down $4$

This order matches $x\mapsto x/2+1$ and $y\mapsto3y-4$.

Example 2

Transform an asymptote

A reciprocal graph

Describe $y=\dfrac{-2}{x+1}+3$ from $y=1/x$.

  • $x+1$ shifts the graph left $1$.
  • The factor $-2$ reflects it in the $x$-axis and stretches vertically by $2$.
  • $+3$ shifts it up $3$.

The asymptotes become $x=-1$ and $y=3$.

Checkpoint

Which rule reflects $y=f(x)$ in the $y$-axis?
Show explanation

Replacing $x$ by $-x$ reverses horizontal coordinates.

Example 3

Recover an equation

Match the vertex and width

A transformed parabola has vertex $(-3,2)$, opens downward, and is twice as steep as $y=x^2$.

Vertex form gives $y=a(x+3)^2+2$. Opening downward makes $a<0$, and the vertical stretch gives $|a|=2$.

$$y=-2(x+3)^2+2.$$

Substituting $x=-3$ confirms the vertex output is $2$.

In the exam

Quick check

Five questions to check the main decisions from this lesson.

Your score is not saved.

  1. $f(x)-5$ moves the graph…
    Show explanation

    Subtracting outside the function lowers every output.

  2. $f(3x)$ has which horizontal scale factor?
    Show explanation

    Inputs reach the old value three times sooner.

  3. Which rule reflects in the $x$-axis?
    Show explanation

    Negating outputs reflects vertically.

  4. The vertical asymptote of $1/(x-4)$ is…
    Show explanation

    The denominator is zero at $x=4$.

  5. A point $(2,-1)$ on $f$ maps under $y=2f(x)+3$ to…
    Show explanation

    The input is unchanged and the output becomes $2(-1)+3=1$.

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