Transformations of Graphs: Shifts, Stretches and Reflections

Q: Why does f(x − 3) move the graph to the right?

The new function reaches each of the parent's values 3 units later. For example, the parent's value at 0 now happens at x = 3.

Q: What is a parent function?

It is the simplest member of a family of graphs, like y = x^2 for parabolas or y = |x| for V shapes. Every other member is a transformation of it.

How one parent graph gives a whole family of graphs, why $f(x - 3)$ moves right rather than left, and how to combine shifts, stretches and reflections, with an explorer to try them live.

3 min read Supports 4 lessons

Learn the shape of a handful of basic graphs, called parent graphs, and you can sketch thousands more. Every graph in the family $y = a\,f(x - h) + k$ is the parent $y = f(x)$ moved and reshaped.

Try each slider on its own first, then combine them. You can switch the parent graph too.

Transformations

y = a\,f(x - h) + k

$h$ shifts right, $k$ shifts up, $|a|$ stretches vertically, and $a < 0$ reflects in the $x$ -axis.

Shifting up and down

Adding $k$ to every output raises every point of the graph by $k$ . So $y = x^2 + 3$ is the parabola $y = x^2$ lifted $3$ units.

Shifting left and right

$y = f(x - 3)$ moves the graph $3$ units to the right, which feels backwards. Here is why: the new graph reaches the parent's old height $3$ steps later. To get the output the parent had at $x = 0$ , you now need $x = 3$ , since $3 - 3 = 0$ .

$(x - 3)^2$ is the parabola moved right by 3.

Watch out

Inside the bracket the direction is reversed: $f(x - 3)$ moves right, $f(x + 3)$ moves left.

Stretches and reflections

Multiplying the output by $a$ stretches the graph away from the $x$ -axis when $|a| > 1$ and squashes it when $|a| < 1$ . If $a$ is negative, the graph also flips upside down.

Putting it together

$y = x^2$
Start from the parent.

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Tip

Do stretches and reflections before the vertical shift. Shifting first and then stretching would stretch the shift as well.

Key idea

This is why vertex form $y = a(x - h)^2 + k$ shows a parabola's turning point at $(h, k)$ , and why $y = A\sin(B(x - h)) + k$ describes every sine wave.

Common questions

Why does f(x − 3) move the graph to the right?: The new function reaches each of the parent's values $3$ units later. For example, the parent's value at $0$ now happens at $x = 3$ .
What is a parent function?: It is the simplest member of a family of graphs, like $y = x^2$ for parabolas or $y = |x|$ for V shapes. Every other member is a transformation of it.

Practise graph transformations

The Transforming Graphs lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Why does f(x − 3) move the graph to the right?

The new function reaches each of the parent's values $3$ units later. For example, the parent's value at $0$ now happens at $x = 3$ .

What is a parent function?

It is the simplest member of a family of graphs, like $y = x^2$ for parabolas or $y = |x|$ for V shapes. Every other member is a transformation of it.