GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Translations and reflections of functions

Swap x for x + 5 in y = f(x). Almost everyone expects the graph to slide 5 right. It slides 5 left, and by the end you’ll know why.

Turn up a in f(x + a). Which way does the graph go?

a in f(x + a) 0

a in f(x + a): 0. x-coordinate of the turning point: -2

This is y = f(x + a) for f(x) = x² + 4x − 12. At a = 0 it is the original graph: roots at −6 and 2, turning point (−2, −16). Before you move the slider, guess: when a goes up, does the parabola go right or left?

Exam line: Set a = −7 for y = f(x − 7): the roots are now 1 and 9 and the turning point is (5, −16). Minus 7 inside the bracket, 7 to the right.

Watch out: Set a = 5. That is y = f(x + 5), and the graph has gone 5 units LEFT, not right.

Why it goes backwards

Reason it through

Why does adding to x inside the bracket move the graph the opposite way?

Link 1 of 4

First link · your turn

Take f(x) = 2x + 1. In f(x + 1), what happens to x before f gets to work on it?

Locked — reveal the link above first

Maths · Algebra

Finding the equation of the moved graph

Setting a = −7 above gave y = f(x − 7), sitting 7 to the right. Here is its equation, line by line, and then the easier outside-the-bracket case.

Goalf(x) = x² + 4x − 12. Find f(x − 7).

f(x) = x² + 4x − 12

Start from the function you are given.

Step 1 of 5

Start from the function you are given.

Watch out: Put the brackets in when you substitute. (x − 7)² is x² − 14x + 49, not x² − 49.

Two minus signs, two different jobs

Which one is the reflection in the y-axis?

You have the graph of y = f(x) and you want its reflection in the y-axis.

Which is closest to what you think right now?

y = −f(x). The minus sign flips the graph, so that’s the reflection in the y-axis.y = f(−x). The minus sign is on the x, so it’s the x-coordinates that flip.Either. −f(x) and f(−x) are the same thing written two ways.

How sure are you?

I'm sureI think soI'm guessing

y = −f(x). The minus sign flips the graph, so that’s the reflection in the y-axis.

This is a common misconception

It does flip the graph, but over the other axis. The minus sign is outside the bracket, so it multiplies the OUTPUT by −1: each point (x, y) becomes (x, −y). The y-coordinates change sign, so the graph turns upside down over the x-axis. (1, 3) becomes (1, −3).

y = f(−x). The minus sign is on the x, so it’s the x-coordinates that flip.

This matches the evidence

Yes. The −1 multiplies the input before f acts, so the same output now comes from the opposite input: (x, y) becomes (−x, y). For f(x) = (x − 3)², the output 0 comes from x = 3, but for f(−x) it comes from x = −3.

Either. −f(x) and f(−x) are the same thing written two ways.

This is a common misconception

They look alike, but the minus sign lands in a different place. In −f(x) it changes the output, so y flips; in f(−x) it changes the input, so x flips. Follow (1, 3): −f(x) sends it to (1, −3) and f(−x) sends it to (−1, 3). Two different points, so two different graphs.

Minus outside the bracket flips y: reflection in the x-axis. Minus inside the bracket flips x: reflection in the y-axis.

Your turn to sketch

Sketch the two reflections, one feature at a time

f(x) = x² + 8x + 12. Find the key features of y = −f(x) and y = f(−x) so you can sketch both.

Key features of y = f(x): roots x = −2 and x = −6, turning point (−4, −4), y-intercept (0, 12).Factorise: (x + 2)(x + 6). Complete the square: (x + 4)² − 4. Put x = 0: 12.
missing step

Which line is step 2?

y = −f(x): the roots x = −2 and x = −6 stay where they are.y = −f(x): the roots move to x = 2 and x = 6.y = −f(x): the roots are no longer roots.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Change the input or the output of f(x), and the whole graph moves. One of the four moves goes the opposite way to what you’d expect.

What you need to know

You can transform a graph without knowing what f is: move each point on y = f(x).
y = f(x) + a is a translation of a in the y direction (up if a is positive, down if negative): (x, y) → (x, y + a).
y = f(x + a) is a translation of −a in the x direction, the opposite way to the sign: (x, y) → (x − a, y). f(x + 3) moves 3 left; f(x − 3) moves 3 right.
y = −f(x) is a reflection in the x-axis: (x, y) → (x, −y).
y = f(−x) is a reflection in the y-axis: (x, y) → (−x, y).
An invariant point is one the transformation doesn’t move: the roots under −f(x), and the y-intercept under f(−x).

The big picture

You can move the graph of y = f(x) without knowing what f is, by moving every point. Changes outside the bracket act on the output: f(x) + a moves the graph a up, and −f(x) reflects it in the x-axis. Changes inside the bracket act on the input and work in reverse: f(x + a) moves the graph a to the left, and f(−x) reflects it in the y-axis. To sketch a transformed quadratic, find its y-intercept, roots and turning point, then move each one.

Key points

1Ask one question first: does the change act on the input (inside the bracket) or the output (outside)?

2Output changes do what they say: + a lifts the graph by a, and the minus sign turns it upside down.

3Input changes work in reverse, because each output now comes from a different input.

4Under f(x) + a the roots stop being roots; under f(x + a) the y-intercept leaves the y-axis; under −f(x) a minimum becomes a maximum.

5Sketching a transformed quadratic: find the y-intercept (x = 0), the roots (factorise) and the turning point (complete the square), then move each feature.

6The equation of a translated graph comes from substituting: replace every x with (x + a), then expand and simplify.

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Worked example

Problem

f(x) = x² + 6x + 5. Sketch y = f(x) + 2, giving the y-intercept, the turning point, and where the old roots end up.

⚠ Watch out

Moving y = f(x + 5) five units to the right. The + 5 is inside the bracket, so it acts on the input and the graph moves the opposite way: 5 units to the left. It is y = f(x − 5) that moves 5 to the right.

🧠

Memory hook

Outside the bracket, as you’d expect. Inside the bracket, in reverse. And the minus sign flips whichever coordinate it touches: outside flips y, inside flips x.

✓

Check yourself

(3, −2) is on y = f(x). Where does it go on y = f(x) − 4, f(x + 2), −f(x) and f(−x)? Answers: (3, −6), (1, −2), (3, 2), (−3, −2).

Flashcards

(15)

Describe the translation that takes y = f(x) to y = f(x) + a.

Translates it by a in the y direction: up for positive a, down for negative a. (x, y) → (x, y + a).

In which direction, and how far, does y = f(x + a) move the graph of y = f(x)?

Translates it by −a in the x direction, the opposite way to the sign of a. (x, y) → (x − a, y).

Which way does y = f(x − 3) move the graph?

3 units to the right (the positive x direction).

Why does f(x + a) move the graph the “wrong” way?

a is added to the input before f acts, so each output now comes from an input a smaller.

Where does the point (x, y) go under y = −f(x)?

To (x, −y): the graph is reflected in the x-axis.

Which axis is y = f(−x) a reflection in, and why?

The y-axis. The input is multiplied by −1 before f acts, so (x, y) → (−x, y).

What is an invariant point?

A point on the graph that the transformation does not move.

Which points are invariant under y = −f(x)?

The points where the graph meets the x-axis, because an output of 0 multiplied by −1 is still 0.

Which point is invariant under y = f(−x)?

The point where the graph meets the y-axis, because an input of 0 is unchanged.

Under y = f(x) + a (a ≠ 0), what happens to the roots of y = f(x)?

They move a units up or down, off the x-axis, so they are no longer roots.

Under y = f(x + a) (a ≠ 0), what happens to the y-intercept of y = f(x)?

It moves a units sideways, off the y-axis. The roots stay roots and the turning point stays a turning point.

What happens to a minimum under y = −f(x)? Under y = f(−x)?

Under −f(x) it becomes a maximum. Under f(−x) it is still a minimum.

Before sketching a transformed quadratic, which three features do you find?

The y-intercept (put x = 0), the roots (factorise) and the turning point (complete the square).

How do you find the equation of y = f(x + a) from f(x)?

Replace every x in f(x) with (x + a), then expand and simplify.

y = cos x looks unchanged under f(−x). Is every point invariant?

No. Only the y-intercept is. (360, 1) moves to (−360, 1), which just happens to be on the curve too.

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.

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Translations and reflections of functions

Turn up a in f(x + a). Which way does the graph go?

Finding the equation of the moved graph

What does each transformation keep?

Which one is the reflection in the y-axis?

Sketch the two reflections, one feature at a time

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