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)(2,\ 5) lies on the curve y=f(x)y = f(x). Write down the coordinates of the corresponding point on the curve y=f(x)+3y = f(x) + 3.

Worked solution

  1. Identify the transformation

    y=f(x)+3y = f(x) + 3

    The +3 is written outside the function, so it changes only the yy-coordinates: the whole curve slides 3 units up. The xx-coordinate of every point stays where it is.

  2. Apply the rule to the point

    (2, 5)→(2, 5+3)(2,\ 5) \to (2,\ 5 + 3)

    Keep the xx-coordinate as 2 and add 3 to the yy-coordinate.

  3. State the image

    (2, 8)(2,\ 8)

    The point (2, 5) on y=f(x)y = f(x) maps to (2, 8). The arrow on the diagram shows the move.

Answer
(2,8)(2, 8)
Question 2
1 markeasy
The point (4, −1)(4,\ -1) lies on the curve y=f(x)y = f(x). Write down the coordinates of the corresponding point on the curve y=f(x)−6y = f(x) - 6.

Worked solution

  1. Identify the transformation

    y=f(x)−6y = f(x) - 6

    The -6 is written outside the function, so it changes only the yy-coordinates: the whole curve slides 6 units down. The xx-coordinate of every point stays where it is.

  2. Apply the rule to the point

    (4, −1)→(4, −1−6)(4,\ -1) \to (4,\ -1 - 6)

    Keep the xx-coordinate as 4 and subtract 6 to the yy-coordinate.

  3. State the image

    (4, −7)(4,\ -7)

    The point (4, -1) on y=f(x)y = f(x) maps to (4, -7). The arrow on the diagram shows the move.

Answer
(4,−7)(4, -7)
Question 3
2 markseasy
The point (3, 7)(3,\ 7) lies on the curve y=f(x)y = f(x). Write down the coordinates of the corresponding point on the curve y=f(x+2)y = f(x + 2).

Worked solution

  1. Identify the transformation

    y=f(x+2)y = f(x + 2)

    The number is inside the bracket with the xx, so this is a horizontal translation. y=f(x+2)y = f(x + 2) moves the curve 2 units to the LEFT — the opposite of the sign you can see. The yy-coordinate does not change.

  2. Apply the rule to the point

    (3, 7)→(3−2, 7)(3,\ 7) \to (3 - 2,\ 7)

    Subtract 2 from the xx-coordinate (the curve moves 2 to the left); leave the yy-coordinate alone.

  3. State the image

    (1, 7)(1,\ 7)

    The point (3, 7) on y=f(x)y = f(x) maps to (1, 7). The arrow on the diagram shows the move.

Answer
(1,7)(1, 7)
Question 4
2 markseasy
The point (−2, 4)(-2,\ 4) lies on the curve y=f(x)y = f(x). Write down the coordinates of the corresponding point on the curve y=f(x−5)y = f(x - 5).

Worked solution

  1. Identify the transformation

    y=f(x−5)y = f(x - 5)

    The number is inside the bracket with the xx, so this is a horizontal translation. y=f(x−5)y = f(x - 5) moves the curve 5 units to the RIGHT — the opposite of the sign you can see. The yy-coordinate does not change.

  2. Apply the rule to the point

    (−2, 4)→(−2+5, 4)(-2,\ 4) \to (-2 + 5,\ 4)

    Add 5 from the xx-coordinate (the curve moves 5 to the right); leave the yy-coordinate alone.

  3. State the image

    (3, 4)(3,\ 4)

    The point (-2, 4) on y=f(x)y = f(x) maps to (3, 4). The arrow on the diagram shows the move.

Answer
(3,4)(3, 4)
Question 5
1 markeasy
The point (6, 2)(6,\ 2) lies on the curve y=f(x)y = f(x). Write down the coordinates of the corresponding point on the curve y=−f(x)y = -f(x).

Worked solution

  1. Identify the transformation

    y=−f(x)y = -f(x)

    The minus sign is outside the function, so every yy-coordinate changes sign: this is a reflection in the xx-axis. The xx-coordinates are unchanged.

  2. Apply the rule to the point

    (6, 2)→(6, −(2))(6,\ 2) \to (6,\ -(2))

    Keep the xx-coordinate and change the sign of the yy-coordinate.

  3. State the image

    (6, −2)(6,\ -2)

    The point (6, 2) on y=f(x)y = f(x) maps to (6, -2). The arrow on the diagram shows the move.

Answer
(6,−2)(6, -2)

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