Transformations of graphs Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Transformations of graphs questions. See exactly how to solve problems on vertical translation, image of a point, horizontal translation, reflection in the x-axis.

vertical translationimage of a pointhorizontal translationreflection in the x-axisreflection in the y-axisdescribing a transformation

GCSE Higher70 questionsStep-by-step solutions

Question 1

1 markeasy

The point

(2,\ 5)

lies on the curve

y = f(x)

. Write down the coordinates of the corresponding point on the curve

y = f(x) + 3

Worked solution

Identify the transformation
$y = f(x) + 3$
The + $3$ is written outside the function, so it changes only the $y$ -coordinates: the whole curve slides $3$ units up. The $x$ -coordinate of every point stays where it is.
Apply the rule to the point
$(2,\ 5) \to (2,\ 5 + 3)$
Keep the $x$ -coordinate as $2$ and add $3$ to the $y$ -coordinate.
State the image
$(2,\ 8)$
The point ( $2$ , $5$ ) on $y = f(x)$ maps to ( $2$ , $8$ ). The arrow on the diagram shows the move.

Answer

(2, 8)

Question 2

1 markeasy

The point

(4,\ -1)

lies on the curve

y = f(x)

. Write down the coordinates of the corresponding point on the curve

y = f(x) - 6

Worked solution

Identify the transformation
$y = f(x) - 6$
The $-6$ is written outside the function, so it changes only the $y$ -coordinates: the whole curve slides $6$ units down. The $x$ -coordinate of every point stays where it is.
Apply the rule to the point
$(4,\ -1) \to (4,\ -1 - 6)$
Keep the $x$ -coordinate as $4$ and subtract $6$ to the $y$ -coordinate.
State the image
$(4,\ -7)$
The point ( $4$ , $-1$ ) on $y = f(x)$ maps to ( $4$ , $-7$ ). The arrow on the diagram shows the move.

Answer

(4, -7)

Question 3

2 markseasy

The point

(3,\ 7)

lies on the curve

y = f(x)

. Write down the coordinates of the corresponding point on the curve

y = f(x + 2)

Worked solution

Identify the transformation
$y = f(x + 2)$
The number is inside the bracket with the $x$ , so this is a horizontal translation. $y = f(x + 2)$ moves the curve $2$ units to the LEFT — the opposite of the sign you can see. The $y$ -coordinate does not change.
Apply the rule to the point
$(3,\ 7) \to (3 - 2,\ 7)$
Subtract $2$ from the $x$ -coordinate (the curve moves $2$ to the left); leave the $y$ -coordinate alone.
State the image
$(1,\ 7)$
The point ( $3$ , $7$ ) on $y = f(x)$ maps to ( $1$ , $7$ ). The arrow on the diagram shows the move.

Answer

(1, 7)

Question 4

2 markseasy

The point

(-2,\ 4)

lies on the curve

y = f(x)

. Write down the coordinates of the corresponding point on the curve

y = f(x - 5)

Worked solution

Identify the transformation
$y = f(x - 5)$
The number is inside the bracket with the $x$ , so this is a horizontal translation. $y = f(x - 5)$ moves the curve $5$ units to the RIGHT — the opposite of the sign you can see. The $y$ -coordinate does not change.
Apply the rule to the point
$(-2,\ 4) \to (-2 + 5,\ 4)$
Add $5$ from the $x$ -coordinate (the curve moves $5$ to the right); leave the $y$ -coordinate alone.
State the image
$(3,\ 4)$
The point ( $-2$ , $4$ ) on $y = f(x)$ maps to ( $3$ , $4$ ). The arrow on the diagram shows the move.

Answer

(3, 4)

Question 5

1 markeasy

The point

(6,\ 2)

lies on the curve

y = f(x)

. Write down the coordinates of the corresponding point on the curve

y = -f(x)

Worked solution

Identify the transformation
$y = -f(x)$
The minus sign is outside the function, so every $y$ -coordinate changes sign: this is a reflection in the $x$ -axis. The $x$ -coordinates are unchanged.
Apply the rule to the point
$(6,\ 2) \to (6,\ -(2))$
Keep the $x$ -coordinate and change the sign of the $y$ -coordinate.
State the image
$(6,\ -2)$
The point ( $6$ , $2$ ) on $y = f(x)$ maps to ( $6$ , $-2$ ). The arrow on the diagram shows the move.

Answer

(6, -2)

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