What a graph transformation does

A graph transformation is a rule that changes a graph’s position, orientation, or scale while leaving enough of its shape intact to recognise the original relationship. We begin with a familiar graph

From there, we predict the graph of a related function rather than plotting it from scratch.

Keep two families separate. A change outside the function, such as $f(x)+b$, acts vertically because it alters the output values. A change inside the function, such as $f(x-a)$, acts horizontally by changing the input value fed into $f$.

A translation moves every point of a graph through the same directed distance. A reflection flips every point across a fixed mirror line. A stretch multiplies distances from a fixed axis by a constant scale factor. When that scale factor lies between $0$ and $1$, the stretch is often called a compression.

The safest way to understand a transformation is to track one point. Suppose $(u,v)$ lies on $y=f(x)$, with input $u$ and output $v$. We can describe the transformed graph by following where this point moves.

Vertical transformations: outside the function

In

Every output increases by $b$, so each point $(u,v)$ on $y=f(x)$ moves to $(u,v+b)$. The graph moves up if $b>0$ and down if $b<0$. Written as a vector, the translation is

$$\begin{pmatrix}0\\ b\end{pmatrix}$$

For $y=-f(x)$, every output changes sign. Thus, $(u,v)$ moves to $(u,-v)$ and the graph is reflected in the $x$-axis. Any point already on the $x$-axis remains fixed, which is why the $x$-axis acts as the mirror line.

For

Multiplying every output by $p$ sends $(u,v)$ to $(u,pv)$. When $p>1$, distances from the $x$-axis increase. When $0<p<1$, those distances decrease.

Take $y=4f(x)+2$. First multiply the old output values by $4$, then add $2$, so $(u,v)$ moves to $(u,4v+2)$. In transformation language, this is a vertical stretch by scale factor $4$, followed by a vertical translation up by $2$.

Horizontal transformations: inside the function

For

Each point $(u,v)$ on $y=f(x)$ moves to $(u+a,v)$. If $a>0$, the graph shifts right; if $a<0$, it shifts left. The translation vector is

$$\begin{pmatrix}a\\0\end{pmatrix}$$

This is where students often get caught: changes inside the function seem to work backwards. The graph of $y=f(x-3)$ moves right by $3$, not left, because the new input $x=3$ produces the old input $0$.

In $y=f(-x)$, the input changes sign. The point $(u,v)$ moves to $(-u,v)$, reflecting the graph in the $y$-axis. Points on the $y$-axis stay where they are.

For

Its horizontal scale factor is $\frac{1}{q}$, so $(u,v)$ moves to $\left(\frac{u}{q},v\right)$. If $q>1$, the graph is compressed horizontally. If $0<q<1$, it is stretched horizontally.

Say the “reciprocal factor” aloud if needed: $y=f(2x)$ has horizontal scale factor $\frac{1}{2}$, whereas $y=f\left(\frac{x}{2}\right)$ has horizontal scale factor $2$.

The standard transformations in one place

The table below is the practical toolkit. Learn the transformations by tracking point movement rather than memorising isolated phrases.

Summary of standard graph transformations.

Here’s a useful split: outside transformations directly change the $y$-coordinates. Inside transformations change the $x$-coordinates in the opposite-looking direction. That is why $f(x)+2$ moves up $2$, while $f(x+2)$ moves left $2$.

Composite transformations and order

A composite transformation applies two or more transformations one after another. Order matters when both transformations act in the same direction and one is a translation.

Start with $y=f(x)$. A vertical stretch by scale factor $2$, followed by a vertical translation up $5$, produces

$$y=2f(x)+5$$

Reverse the order—a vertical translation up $5$, then a vertical stretch by scale factor $2$—and the result is

$$y=2(f(x)+5)=2f(x)+10$$

These graphs aren’t the same. In the second case, the translation happens before the stretch, so the graph is lifted twice as far.

If one transformation is horizontal and the other vertical, changing the order does not change the final graph. For instance, shifting right before stretching vertically gives the same result as stretching vertically before shifting right. A stretch and reflection in the same direction can also be combined in either order. Thus, $y=-3f(x)$ represents a reflection in the $x$-axis together with a vertical stretch by scale factor $3$.

Horizontal composite transformations need extra care. Consider $y=f\left(\frac{1}{3}(x+2)\right)$, where the expression inside the function is grouped. Inside the input expression, $x$ is first shifted left by $2$ and then multiplied by $\frac{1}{3}$. Tracking a point shows the visible graph effect most clearly: a horizontal stretch by scale factor $3$, followed by a translation left by $2$. A different intended order would give different algebra, such as $y=f\left(\frac{1}{3}x+2\right)$.

The safest approach is to pick a simple point on the original graph and follow its coordinates through each stated transformation. This avoids the classic wrong-direction answer for expressions such as $f(x-3)$.

Transforming common functions

Transformations link directly to function families studied earlier. Consider a quadratic in vertex form

This is a transformed version of $y=x^2$, with vertex $(h,k)$. It opens upwards if $a>0$. If $a<0$, the negative sign includes a reflection in the $x$-axis.

Completing the square, for example, rewrites an expanded quadratic in a form that shows its transformations. If

$$y=5x^2+10x+9$$

then

$$\begin{aligned} y&=5(x^2+2x)+9\\ &=5(x+1)^2+4. \end{aligned}$$

Starting from $y=x^2$, the graph is stretched vertically by scale factor $5$, translated left by $1$, and translated up by $4$. Its vertex is $(-1,4)$.

Transformations also help us read rational functions. The reciprocal graph $y=\frac{1}{x}$ has vertical asymptote $x=0$ and horizontal asymptote $y=0$. For the transformed function

$$y=3-\frac{5}{x+1}$$

the graph is translated left by $1$, reflected in the $x$-axis, stretched vertically by scale factor $5$, and translated up by $3$. Its vertical asymptote is $x=-1$, while its horizontal asymptote is $y=3$.

Transformations are a representation tool, not merely a sketching trick. Writing the same function in a suitable symbolic form can reveal key graph features: the vertex of a quadratic, the asymptotes of reciprocal or rational functions, and later the amplitude and period of trigonometric functions.