GCSE Maths revision · Higher: number and algebra

Transforming graphs

Higher only

The idea

$y = f(x) + a$ moves the graph up by $a$ . $y = f(x + a)$ moves it left by $a$ . $y = -f(x)$ reflects it in the $x$ -axis and $y = f(-x)$ reflects it in the $y$ -axis.

Spot where the change is: outside the bracket affects $y$ , inside affects $x$ .
Outside acts as you expect; inside acts the opposite way.
Move key points, such as the turning point, then redraw.

Watch out $f(x + 3)$ moves the graph $3$ to the left, not the right.

See it worked

Worked exampleNon-calculator4 marks

The curve $y = f(x)$ has a turning point at $(2, -3)$ .

(a) Write down the coordinates of the turning point of $y = f(x) + 4$ .

(b) Write down the coordinates of the turning point of $y = f(x - 3)$ .

(c) Write down the coordinates of the turning point of $y = -f(x)$ .

(d) Write down the coordinates of the turning point of $y = f(-x)$ .

(a) $+4$ outside the function moves the curve up 4: $(2, -3 + 4) = (2, 1)$ .
(b) $x - 3$ inside the function moves the curve 3 to the right: $(5, -3)$ .
(c) $-f(x)$ reflects the curve in the $x$ -axis: $(2, 3)$ .
(d) $f(-x)$ reflects the curve in the $y$ -axis: $(-2, -3)$ .

Answer (a) $(2, 1)$ ; (b) $(5, -3)$ ; (c) $(2, 3)$ ; (d) $(-2, -3)$

Your turn

Question 1Calculator allowed1 mark

Which transformation maps the graph y = f(x) onto the graph y = f(x $-$ 3)?

Question 2Calculator allowed3 marks

f(x) = $x^{2}$ $-$ 6x + 5. Find the coordinates of the maximum point on the graph y = $-f$ (x $-$ 2) + 7.

Question 3Non-calculator4 marks

The graph $y=f(x)$ has its unique minimum at $(4,1)$ .

(a) Find the minimum point of $y=f(-x)$ .

(b) Find the minimum point of $y=f(-x)-6$ .

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