Transformations of graphs Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Transformations of graphs questions. See exactly how to solve problems on vertical translation, f(x)+a, image of a point, f(x)-a.

vertical translationf(x)+aimage of a pointf(x)-ahorizontal translationf(x-a)
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The point P(2, 5)P(2,\ 5) lies on the curve y=f(x)y=f(x). Write down the coordinates of the image of PP on the curve y=f(x)+3y=f(x)+3.

Worked solution

  1. Recall the transformation rule

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

    Adding 3 to the whole function moves every point up by 3, so the y-coordinate increases by 3.

  2. Write down the original point

    P(2, 5)P(2,\ 5)

    P has coordinates (2,\ 5) and lies on the curve y=f(x).

  3. Apply the rule to the coordinates

    (2, 5) ⟶ (2, 8)(2,\ 5)\ \longrightarrow\ (2,\ 8)

    Only change the coordinate that this transformation affects; the other coordinate stays exactly the same. Remember from plotting graphs that 'up' means the y-value grows.

  4. State the image

    P′(2, 8)P'(2,\ 8)

    So this is the position of the point after the transformation.

Answer
(2, 8)(2,\ 8)
Question 2
2 markseasy
The point P(−1, 6)P(-1,\ 6) lies on y=f(x)y=f(x). State the coordinates of the corresponding point on y=f(x)−4y=f(x)-4.

Worked solution

  1. Recall the transformation rule

    y=f(x)−4y=f(x)-4

    Subtracting 4 from the whole function moves every point down by 4, so the y-coordinate decreases by 4.

  2. Write down the original point

    P(−1, 6)P(-1,\ 6)

    P has coordinates (-1,\ 6) and lies on the curve y=f(x).

  3. Apply the rule to the coordinates

    (−1, 6) ⟶ (−1, 2)(-1,\ 6)\ \longrightarrow\ (-1,\ 2)

    Only change the coordinate that this transformation affects; the other coordinate stays exactly the same.

  4. State the image

    P′(−1, 2)P'(-1,\ 2)

    So this is the position of the point after the transformation.

Answer
(−1, 2)(-1,\ 2)
Question 3
2 markseasy
The point P(3, 7)P(3,\ 7) lies on y=f(x)y=f(x). Find the image of PP under the transformation to y=f(x−2)y=f(x-2).

Worked solution

  1. Recall the transformation rule

    y=f(x−2) ⇒ translation (20)y=f(x-2)\ \Rightarrow\ \text{translation}\ \binom{2}{0}

    Replacing x with x-2 moves the curve 2 to the right, so the x-coordinate increases by 2.

  2. Write down the original point

    P(3, 7)P(3,\ 7)

    P has coordinates (3,\ 7) and lies on the curve y=f(x).

  3. Apply the rule to the coordinates

    (3, 7) ⟶ (5, 7)(3,\ 7)\ \longrightarrow\ (5,\ 7)

    Only change the coordinate that this transformation affects; the other coordinate stays exactly the same. Inside-the-bracket changes act on x and go the 'opposite' way to the sign.

  4. State the image

    P′(5, 7)P'(5,\ 7)

    So this is the position of the point after the transformation.

Answer
(5, 7)(5,\ 7)
Question 4
2 markseasy
The point P(4, −2)P(4,\ -2) lies on y=f(x)y=f(x). Write down the coordinates of its image on y=f(x+5)y=f(x+5).

Worked solution

  1. Recall the transformation rule

    y=f(x+5) ⇒ translation (−50)y=f(x+5)\ \Rightarrow\ \text{translation}\ \binom{-5}{0}

    Replacing x with x+5 moves the curve 5 to the left, so the x-coordinate decreases by 5.

  2. Write down the original point

    P(4, −2)P(4,\ -2)

    P has coordinates (4,\ -2) and lies on the curve y=f(x).

  3. Apply the rule to the coordinates

    (4, −2) ⟶ (−1, −2)(4,\ -2)\ \longrightarrow\ (-1,\ -2)

    Only change the coordinate that this transformation affects; the other coordinate stays exactly the same.

  4. State the image

    P′(−1, −2)P'(-1,\ -2)

    So this is the position of the point after the transformation.

Answer
(−1, −2)(-1,\ -2)
Question 5
2 markseasy
The point P(3, 4)P(3,\ 4) lies on y=f(x)y=f(x). State the coordinates of the image of PP on y=2f(x)y=2f(x).

Worked solution

  1. Recall the transformation rule

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

    Multiplying the whole function by 2 doubles every y-coordinate: a vertical stretch, scale factor 2.

  2. Write down the original point

    P(3, 4)P(3,\ 4)

    P has coordinates (3,\ 4) and lies on the curve y=f(x).

  3. Apply the rule to the coordinates

    (3, 4) ⟶ (3, 8)(3,\ 4)\ \longrightarrow\ (3,\ 8)

    Only change the coordinate that this transformation affects; the other coordinate stays exactly the same.

  4. State the image

    P′(3, 8)P'(3,\ 8)

    So this is the position of the point after the transformation.

Answer
(3, 8)(3,\ 8)

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