Function Transformations

Precalculus is the course that gets functions ready for calculus. A function is a rule with an input and one output, and here you learn how to graph, change, and reverse them.

Function Transformations: the central idea

In $y=a f(b(x-h))+k$, outside changes act vertically while inside changes act horizontally and in the inverse scale or direction.

Words you need

Parent function

A parent function is the simplest form of a family of graphs, such as $y=x^2$ for parabolas or $y=|x|$ for V shapes, before any shifting or stretching happens.

Translation

A translation is a slide that moves every point of a graph the same distance in the same direction without changing its size or shape.

Reflection

A reflection is a flip of the graph across a line, so a minus sign outside flips it over the $x$-axis and a minus sign inside flips it over the $y$-axis.

Vertical stretch

A vertical stretch multiplies every output by a number bigger than $1$, pulling the graph away from the $x$-axis and making it look steeper.

Horizontal compression

A horizontal compression multiplies the input by a number bigger than $1$, pushing the graph toward the $y$-axis so it looks narrower.

Order of transformations

The order of transformations matters because stretching then shifting lands in a different place than shifting then stretching, so inside changes are handled first, from the input outward.

What to know before this lesson

Recognize parent functions, read function notation, and plot key points, intercepts, and asymptotes.

If one of those prerequisites is uncertain, use the Precalculus subject guide to locate the earlier concept before memorizing a procedure.

Function transformations: a worked example

Every step, with the arithmetic

1. Step 1 - Name the parent and its anchor pointsParent $y=x^2$ with points $(-1,1)$, $(0,0)$, $(1,1)$, and $(2,4)$
2. Step 2 - Write the new function$g(x)=-3(x-2)^2+5$
3. Step 3 - Read the four changesInside $(x-2)$ shifts right $2$; the $3$ stretches heights by $3$; the minus sign flips it upside down; the $+5$ lifts it $5$
4. Step 4 - Build the mapping rule$(u,\,u^2)\;\to\;(u+2,\;-3u^2+5)$
5. Step 5 - Move the vertex$u=0$: $(0+2,\;-3(0)+5)=(2,5)$
6. Step 6 - Move a second point$u=1$: $(1+2,\;-3(1)+5)=(3,2)$
7. Step 7 - Move a third point$u=2$: $(2+2,\;-3(4)+5)=(4,-7)$
8. Step 8 - Check one point in the equation$g(3)=-3(3-2)^2+5=-3(1)+5=2$, matching the mapped point $(3,2)$; the range is $y\le5$

$f(x-3)+2$ shifts the graph right $3$ because the old input $x$ occurs when the new input is $x+3$, then raises every output by $2$.

What each change to the equation does to the graph and to a single point (x, y)

The step-by-step method for function transformations

1. Identify the parent function and record several anchor points or defining features.
2. Apply horizontal changes to inputs, then vertical changes to outputs.
3. Move intercepts, asymptotes, domain, and range consistently and test transformed points in the new equation.

How to check your answer

Choose a parent point $(u,f(u))$ and verify that $(u+3,f(u)+2)$ satisfies the transformed equation.

Use intercepts, end behavior, symmetry, and a few exact points to test whether a formula and graph describe the same function.

A mistake that changes the mathematics

The sign inside parentheses appears reversed: $f(x-3)$ moves right, not left.

Where you will actually use this

Sound and audio

A volume knob is a vertical stretch of a sound wave, and a pitch change is a horizontal squeeze, so $\sin(2x)$ sounds one octave above $\sin(x)$.

Tides and daylight hours

Both are modelled as $y=a\sin(b(x-h))+k$, where $k$ sets the average level, $a$ sets how far it swings, $b$ sets how often it repeats, and $h$ sets when the peak arrives.

Image editing

Cropping, resizing, and mirroring a photo apply the same shift, stretch, and reflect rules to every pixel coordinate at once.

How function transformations connects to the rest of precalculus

• Geometric transformations — Translations, reflections, and dilations of a shape on a grid are the same moves, described with coordinate rules instead of function notation.
• Inverse functions — Finding an inverse is itself a transformation: it reflects the whole graph across the line $y=x$.
• The unit circle — Amplitude, period, and phase shift for sine and cosine are exactly the $a$, $b$, and $h$ of this lesson applied to a circular parent function.

Try a transfer problem

Describe every transformation of $y=-2\sqrt{3(x+1)}+4$ and map three parent-function points.

Show the worked answer

Start from the parent $y=\sqrt{x}$ and read $y=-2\sqrt{3(x+1)}+4$ from the inside out. Inside the radical, $(x+1)$ shifts the graph left $1$ unit, and the multiplier $3$ squeezes it horizontally toward the $y$-axis by a factor of $\tfrac13$. Outside, the $2$ stretches every height to double, the minus sign flips the graph upside down across the $x$-axis, and the $+4$ raises everything $4$ units. The full mapping rule is $(u,\sqrt{u})\to\left(\tfrac{u}{3}-1,\;-2\sqrt{u}+4\right)$. Mapping three parent points: $(0,0)\to(-1,4)$, which is the new starting corner; $(9,3)\to(2,-2)$; and $(36,6)\to(11,-8)$. Check the middle one directly: $-2\sqrt{3(2+1)}+4=-2\sqrt{9}+4=-6+4=-2$. The domain is $x\ge-1$ and the range is $y\le4$, since the graph starts at its highest point and falls from there.

Work without copying the example. When finished, use the relevant focused calculator or formula reference to check the setup and result, then correct the first line where your reasoning changed. When the method feels reliable, move to precalculus practice questions.

Questions about function transformations

Why does $f(x-3)$ move the graph right instead of left?

Because the input has to work harder to reach the same value. The parent hits its special output when the thing in the parentheses equals $0$, and $x-3=0$ happens at $x=3$. So the feature that used to sit at $0$ now sits at $3$, which is a move to the right.

If a graph is both stretched and shifted, which do I do first?

Handle the input side before the output side. For $y=2f(3(x-1))+5$, shift right $1$ and squeeze by $\tfrac13$ first, then double the heights and lift by $5$. Doing the vertical shift before the vertical stretch would multiply the shift too and land the graph in the wrong place.

What is the difference between $-f(x)$ and $f(-x)$?

$-f(x)$ negates the output, so the graph flips top-to-bottom over the $x$-axis and $(2,5)$ becomes $(2,-5)$. $f(-x)$ negates the input, so it flips left-to-right over the $y$-axis and $(2,5)$ becomes $(-2,5)$. For an even function like $x^2$, $f(-x)$ changes nothing at all.

How do transformations affect asymptotes and domain?

They move right along with the graph. The vertical asymptote of $y=1/x$ sits at $x=0$, so for $y=1/(x-4)+2$ it slides to $x=4$ and the horizontal asymptote rises to $y=2$. The domain becomes every number except $4$. Whatever you do to the points, do to the boundaries.