GCSE · Maths · AQA · Spec 8300 · Higher

Translations and reflections of functions (Higher)

Put +3 inside a bracket and the graph slides left. It sounds backwards. Once you see why, you won't get it wrong again.

Increase a. Which way does the curve slide?

a in y = f(x + a) 0

a in y = f(x + a): 0. x-coordinate of the turning point: 1

Red line. It runs from where the turning point started, at (1, −4), to where it is now. Push a up to 3 and the red line points 3 units LEFT. Pull a down to −3 and it points 3 units right.

The curve is y = f(x + a). When a = 0 it is y = f(x) itself: a U-shaped curve with its turning point at (1, −4). Before you move a, predict: when a goes up to 3, will the curve slide left or right?

So why does +3 send it left?

The graph of y = f(x) has its lowest point at (1, −4). That means f(1) = −4: the lowest output happens when the bracket holds 1. Now think about y = f(x + 3).

Which is closest to what you think happens to the graph?

It moves 3 to the right. It's a plus, and plus means right along the x-axis.It moves up 3, because you're adding 3.It moves 3 to the left, because every output now happens 3 earlier in x.It changes shape. The numbers inside are bigger, so the curve gets steeper.

How sure are you?

I'm sureI think soI'm guessing

It moves 3 to the right. It's a plus, and plus means right along the x-axis.

This is a common misconception

That's the idea almost everyone starts with, and the curve in the graph above refused to do it. Follow the lowest point. It happens when the bracket holds 1. In f(x + 3) the bracket holds x + 3, so we need x + 3 = 1, which means x = −2. The lowest point is now at x = −2. It has moved from 1 to −2: that's 3 to the LEFT.

It moves up 3, because you're adding 3.

This is a common misconception

Adding 3 to the answer, y = f(x) + 3, would lift every point by 3. But here the 3 is added to x, inside the bracket, before f does anything. The outputs are exactly the same numbers as before: the lowest point is still at a height of −4. What changes is WHERE along the x-axis each height happens.

It moves 3 to the left, because every output now happens 3 earlier in x.

This matches the evidence

Exactly. Whatever f(x) did at some x-value, f(x + 3) does 3 units sooner, because the bracket gets there 3 early. The lowest point moves from (1, −4) to (−2, −4), every other point shifts left by 3 in the same way, and the shape doesn't change.

It changes shape. The numbers inside are bigger, so the curve gets steeper.

This is a common misconception

Look back at the graph: as you moved a, the curve never got steeper or flatter. Adding a number inside the bracket reaches the same outputs, in the same order, just 3 units earlier. Same shape, new place.

Follow one point. For y = f(x + a), the bracket reaches each old value when x is a smaller than before, so a positive a moves the graph left by a, and a negative a moves it right.

Outside or inside? Name the move

Pick an equation, then the move it makes to the graph of y = f(x). Look first at where the change sits: outside the f, or inside the bracket?

Still to sort

Translation up or down (0)

A number added to, or taken from, the whole of f(x). It changes every y-value by that amount.

Where the line is: The number is outside the f. If it's inside the bracket with x, it belongs in the left-or-right column instead.

Translation left or right (0)

A number added to, or taken from, x inside the bracket. It changes every x-value, the opposite way to its sign.

Where the line is: The number is inside the bracket. The heights of the points don't change at all.

Reflection in the x-axis (0)

A minus sign in front of the whole f(x), as in y = −f(x). Every y-value changes sign, so the graph flips upside down.

Where the line is: The minus is outside the f, so it flips the y-values. Points on the x-axis stay exactly where they are.

Reflection in the y-axis (0)

A minus sign on the x inside the bracket, as in y = f(−x). Every x-value changes sign, so the graph flips left to right.

Where the line is: The minus is inside the bracket, so it flips the x-values. Points on the y-axis stay exactly where they are.

7 of 7 still to sort.

Each equation is a change to the graph of y = f(x). Outside the f means the change happens to the output, y. Inside the bracket means it happens to the input, x.

Sketching by moving key points

Problem

The graph of y = f(x) is an ∩-shaped curve. It crosses the x-axis at (−1, 0) and (3, 0), crosses the y-axis at (0, 3), and has its maximum point at (1, 4). Sketch the graph of y = f(−x).

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Follow one point, and the whole graph follows.

What you need to know

y = f(x) + a translates the graph of y = f(x) up by a (down if a is negative).
y = f(x + a) translates the graph left by a (right if a is negative): the opposite way to the sign.
y = −f(x) reflects the graph in the x-axis; y = f(−x) reflects it in the y-axis.
Translations and reflections never change a graph's shape, only its position or which way round it faces.
To sketch a transformed graph, move the key points (turning points and axis crossings), draw the same shape through them, and label them.

The big picture

If you know the graph of y = f(x), you can sketch four close relatives without plotting a single new point. y = f(x) + a moves the graph up by a. y = f(x + a) moves it left by a, the opposite way to the sign, because every output now happens a units earlier in x. y = −f(x) reflects it in the x-axis, and y = f(−x) reflects it in the y-axis. None of these changes the shape, so to sketch one you move the key points and draw the same shape through them.

Key points

1Outside the f changes y-values; inside the bracket changes x-values.

2Inside the bracket, the graph moves the opposite way to the sign: f(x + 3) moves left 3.

3A minus outside flips the graph upside down; a minus inside flips it left to right.

4Points on a mirror line don't move when you reflect in that line.

5Sketch = move the key points, keep the shape, label the new points.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

The graph of y = f(x) has a minimum point at (−1, −4) and crosses the y-axis at (0, −3). For the graph of y = −f(x), find the turning point and the y-intercept, and say whether the turning point is a maximum or a minimum.

⚠ Watch out

Mixing up y = −f(x) and y = f(−x). Look at where the minus sits. In front of the f, it flips the y-values: a reflection in the x-axis. On the x inside the bracket, it flips the x-values: a reflection in the y-axis.

🧠

Memory hook

Outside tells the truth, inside does the opposite. f(x) + 3 goes up 3, just as it says. f(x + 3) goes left 3, the opposite of what the + suggests.

✓

Check yourself

The graph of y = f(x) has a turning point at (−2, 5). Without drawing anything, write down the turning point of y = f(x) − 3, and then of y = f(x − 3).

Flashcards

(5)

What does y = f(x) + a do to the graph of y = f(x)?

Translates it up by a (down if a is negative). Every y-value changes by a; every x-value stays the same.

For a positive a, which way does y = f(x + a) move the graph of y = f(x)?

Left by a. The bracket reaches each old value when x is a smaller than before, so each output happens a units earlier.

Which reflection is y = −f(x), and which points stay where they are?

A reflection in the x-axis: every y-value changes sign, so the graph turns upside down and a minimum becomes a maximum. Points on the x-axis, where the graph crosses it, don't move.

You reflect the graph of y = f(x) in the y-axis. What is the new equation?

y = f(−x). Each point (x, y) goes to (−x, y), so the graph flips left to right. The y-intercept stays where it is.

How do you sketch a translation or reflection of a given graph?

Name the move, use its rule to move the key points (turning points and axis crossings), then draw the same shape through the new points and label them.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Increase a. Which way does the curve slide?

So why does +3 send it left?

Outside or inside? Name the move

More AQA GCSE Maths topics