Transformations of graphs

What a transformation does

A transformation changes an existing graph $y = f(x)$ in a predictable way — it slides, flips or stretches the whole curve without changing its basic family.
Learn these once and they apply to every graph: parabola, cubic, exponential or hyperbola.

Translations (sliding the graph)

$y = f(x) + k$ — slides the graph up by $k$ (or down if $k$ is negative). It changes the height, so the whole curve moves vertically.
$y = f(x - h)$ — slides the graph right by $h$ $h$ (or left if $h$ $h$ is negative).
- Watch the sign: $f(x - 3)$ moves right 3, $f(x + 3)$ moves left 3 — the shift is the opposite sign to the number with $x$ .

A base parabola with a dashed copy shifted up, labelled f of x plus k, and a dashed copy shifted right, labelled f of x minus h, each with an arrow showing the movement.

Reflections (flipping the graph)

$y = -f(x)$ — reflects the graph in the $x$ -axis (flips it upside down); every $y$ swaps sign.
$y = f(-x)$ — reflects the graph in the $y$ -axis (flips it left–right).

Stretches (scaling the graph)

$y = a\,f(x)$ — a vertical stretch by a factor of $a$ : every $y$ -value is multiplied by $a$ , so the graph is pulled taller (if $a > 1$ ) or squashed (if $0 < a < 1$ ).
This is exactly what the $a$ in $y = ax^2$ did to a parabola — the same idea works for any $f(x)$ .

Combining transformations

A single equation can pack in several transformations. Read them off the standard shape $y = a\,f(x - h) + k$ $y = a f (x - h) + k$ :
- $a$ — vertical stretch / reflection (if negative),
- $h$ — horizontal slide,
- $k$ — vertical slide.

Worked ExampleDescribing a transformation

Describe how the graph of $y = (x - 2)^2 + 3$ is obtained from $y = x^2$ , and state its vertex.

Step 1 — Read the horizontal shift

The " $x - 2$ " means slide the base parabola right by 2.

Step 2 — Read the vertical shift

The " $+ 3$ " means slide it up by 3.

Step 3 — Combine and state the vertex

Starting from $y = x^2$ (vertex at the origin), moving right 2 and up 3 puts the vertex at $(2, 3)$ . The parabola keeps its shape and upward direction.

Worked ExampleA reflection and a shift

The graph of $y = 2^{\,x}$ is reflected in the $x$ -axis and then moved up 5. Write the new equation and give its horizontal asymptote.

Step 1 — Reflect in the $x$ -axis

Reflecting $y = 2^{\,x}$ gives $y = -2^{\,x}$ (every height flips sign).

Step 2 — Move up 5

Adding 5: $y = -2^{\,x} + 5$ .

Step 3 — Asymptote

The original asymptote $y = 0$ also moves up 5, so the new asymptote is $y = 5$ ; the curve now approaches $5$ from below.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Describe the transformation that takes $y = x^2$ to $y = x^2 - 4$ .

Merit

The graph of $y = f(x)$ is translated to $y = f(x + 3) - 2$ . Describe the two moves and state where the point $(0, 0)$ on $f$ ends up.

Excellence

The parabola $y = x^2$ is reflected in the $x$ -axis, stretched vertically by a factor of 2, then moved up 8. Write the resulting equation and find where it crosses the $x$ -axis.

Transformations of graphs

Covers AS91257

What a transformation does

A transformation changes an existing graph $y = f(x)$ in a predictable way — it slides, flips or stretches the whole curve without changing its basic family.
Learn these once and they apply to every graph: parabola, cubic, exponential or hyperbola.

Translations (sliding the graph)

$y = f(x) + k$ — slides the graph up by $k$ (or down if $k$ is negative). It changes the height, so the whole curve moves vertically.
$y = f(x - h)$ — slides the graph right by $h$ $h$ (or left if $h$ $h$ is negative).
- Watch the sign: $f(x - 3)$ moves right 3, $f(x + 3)$ moves left 3 — the shift is the opposite sign to the number with $x$ .

A base parabola with a dashed copy shifted up, labelled f of x plus k, and a dashed copy shifted right, labelled f of x minus h, each with an arrow showing the movement.

Reflections (flipping the graph)

$y = -f(x)$ — reflects the graph in the $x$ -axis (flips it upside down); every $y$ swaps sign.
$y = f(-x)$ — reflects the graph in the $y$ -axis (flips it left–right).

Stretches (scaling the graph)

$y = a\,f(x)$ — a vertical stretch by a factor of $a$ : every $y$ -value is multiplied by $a$ , so the graph is pulled taller (if $a > 1$ ) or squashed (if $0 < a < 1$ ).
This is exactly what the $a$ in $y = ax^2$ did to a parabola — the same idea works for any $f(x)$ .

Combining transformations

A single equation can pack in several transformations. Read them off the standard shape $y = a\,f(x - h) + k$ $y = a f (x - h) + k$ :
- $a$ — vertical stretch / reflection (if negative),
- $h$ — horizontal slide,
- $k$ — vertical slide.

Worked ExampleDescribing a transformation

Describe how the graph of $y = (x - 2)^2 + 3$ is obtained from $y = x^2$ , and state its vertex.

Step 1 — Read the horizontal shift

The " $x - 2$ " means slide the base parabola right by 2.

Step 2 — Read the vertical shift

The " $+ 3$ " means slide it up by 3.

Step 3 — Combine and state the vertex

Starting from $y = x^2$ (vertex at the origin), moving right 2 and up 3 puts the vertex at $(2, 3)$ . The parabola keeps its shape and upward direction.

Worked ExampleA reflection and a shift

The graph of $y = 2^{\,x}$ is reflected in the $x$ -axis and then moved up 5. Write the new equation and give its horizontal asymptote.

Step 1 — Reflect in the $x$ -axis

Reflecting $y = 2^{\,x}$ gives $y = -2^{\,x}$ (every height flips sign).

Step 2 — Move up 5

Adding 5: $y = -2^{\,x} + 5$ .

Step 3 — Asymptote

The original asymptote $y = 0$ also moves up 5, so the new asymptote is $y = 5$ ; the curve now approaches $5$ from below.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Describe the transformation that takes $y = x^2$ to $y = x^2 - 4$ .

Merit

The graph of $y = f(x)$ is translated to $y = f(x + 3) - 2$ . Describe the two moves and state where the point $(0, 0)$ on $f$ ends up.

Excellence

The parabola $y = x^2$ is reflected in the $x$ -axis, stretched vertically by a factor of 2, then moved up 8. Write the resulting equation and find where it crosses the $x$ -axis.

Transformations of graphs

What a transformation does

Translations (sliding the graph)

Reflections (flipping the graph)

Stretches (scaling the graph)

Combining transformations

Step 1 — Read the horizontal shift

Step 2 — Read the vertical shift

Step 3 — Combine and state the vertex

Step 1 — Reflect in the xxx-axis

Step 2 — Move up 5

Step 3 — Asymptote

Test yourself

Practice by grade

Transformations of graphs

What a transformation does

Translations (sliding the graph)

Reflections (flipping the graph)

Stretches (scaling the graph)

Combining transformations

Step 1 — Read the horizontal shift

Step 2 — Read the vertical shift

Step 3 — Combine and state the vertex

Step 1 — Reflect in the xxx-axis

Step 2 — Move up 5

Step 3 — Asymptote

Test yourself

Practice by grade

Step 1 — Reflect in the $x$ -axis

Step 1 — Reflect in the $x$ -axis