Function Transformations: Shift, Stretch & Reflect Explained

Function transformations take a parent graph $y=f(x)$ and build a new graph

$$y=af(x-h)+k.$$

Three numbers do the work: $a$ stretches or flips, $h$ shifts left or right, and $k$ shifts up or down. The live board below freezes the parent $y=x^2$ in gray and the transformed curve in green so you can drag those numbers, not only read them.

What $a$, $h$, and $k$ mean

Start from a parent $y=f(x)$. The transformed graph is $y=af(x-h)+k$.

Same parent shape. Only the three numbers change where it sits and how tall it is.

Notation note. Classroom $a$, $h$, and $k$ are the same moves the handbook writes with $a$ and $b$ on transformation of functions. A shift there is $y=f(x-a)+b$, so handbook $a$ is this post’s $h$ and handbook $b$ is this post’s $k$. A positive stretch there is $y=bf(x/a)$. This post keeps $a$, $h$, and $k$ so the three sliders match what many worksheets print. The handbook page keeps its own letters.

Worked example — from $y=x^2$ to $y=2(x-1{)}^2+3$

Parent: $f(x)=x^2$ (the gray dashed curve).

Target:

$$y=2(x-1{)}^2+3.$$

Read the three numbers: $a=2$, $h=1$, $k=3$.

1. Stretch by $2$: $y=2x^2$ is steeper than $y=x^2$.
2. Shift right by $1$: replace $x$ with $x-1$ → $y=2(x-1{)}^2$.
3. Shift up by $3$: add $3$ → $y=2(x-1{)}^2+3$.

Vertex of the parent is $(0,0)$. After these steps the vertex sits at $(1,3)$. Check one point: at $x=1$, $y=2(0{)}^2+3=3$. At $x=2$, $y=2(1{)}^2+3=5$. Set the sliders to $a=2$, $h=1$, $k=3$ and match that green curve to the algebra — the vertex should sit clear of the slider strip near the top of the board.

Reflect. If $a=-2$ instead, the same shifts give $y=-2(x-1{)}^2+3$: the parabola opens downward (flip over the x-axis) with vertex still at $(1,3)$. Negative $a$ is still the same parameter — stretch size $|a|$, then flip.

Drag the sliders

Gray dashed = parent $y=x^2$. Green = $y=a(x-h{)}^2+k$.

• Drag $a$: height and opening direction change; the vertex’s $x$-coordinate stays put until you move $h$.
• Drag $h$: the whole green curve slides left or right.
• Drag $k$: the whole green curve slides up or down.

That is the check that the figure matches the formula: one parent shape, three numbers, and every green point is the parent rule after stretch/flip and shift. If you already know how to graph a function as pairs $(x,f(x))$, transformations are the next habit — change the rule, keep reading heights. A sine wave is another parent you can transform the same way ($a\sin (x-h)+k$); the wave page and the sine and cosine handbook keep the trig story — this post stays on the $a,h,k$ machine itself.

GoodMaith keeps a relation like this as an object on a live board: name a parent, attach sliders for $a$, $h$, and $k$, and see why $y=af(x-h)+k$ is exactly the stretched, shifted, or flipped copy of $y=f(x)$. The answer was never only the words “function transformations”; it was watching one parent become the green curve under three drags.