Further transformations Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Further transformations questions. See exactly how to solve problems on transformations, image-of-point, describe, range.

transformationsimage-of-pointdescriberangeequationidentify
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
The point (2, 3)\left(2,\ 3\right) lies on the curve y=f(x)y=f(x). Find the coordinates of its image under the transformation y=f(x)+4y=f{\left(x \right)} + 4.

Worked solution

  1. Write down the transformation applied to y=f(x)

    y=f(x)+4y=f{\left(x \right)} + 4

    This equation defines the transformed curve.

  2. Map the given point using the transformation

    (2, 3)↦(2, 7)\left(2,\ 3\right)\mapsto\left(2,\ 7\right)

    Transform the x- and y-coordinates according to the equation.

  3. State the image of the point

    (2, 7)\left(2,\ 7\right)

    This is the required image point.

Answer
(2, 7)\left(2,\ 7\right)
Question 2
2 markseasy
The point (1, 5)\left(1,\ 5\right) lies on the curve y=f(x)y=f(x). Find the coordinates of its image under the transformation y=f(x−3)y=f{\left(x - 3 \right)}.

Worked solution

  1. Write down the transformation applied to y=f(x)

    y=f(x−3)y=f{\left(x - 3 \right)}

    This equation defines the transformed curve.

  2. Map the given point using the transformation

    (1, 5)↦(4, 5)\left(1,\ 5\right)\mapsto\left(4,\ 5\right)

    Transform the x- and y-coordinates according to the equation.

  3. State the image of the point

    (4, 5)\left(4,\ 5\right)

    This is the required image point.

Answer
(4, 5)\left(4,\ 5\right)
Question 3
2 markseasy
The point (4, 2)\left(4,\ 2\right) lies on the curve y=f(x)y=f(x). Find the coordinates of its image under the transformation y=2f(x)y=2 f{\left(x \right)}.

Worked solution

  1. Write down the transformation applied to y=f(x)

    y=2f(x)y=2 f{\left(x \right)}

    This equation defines the transformed curve.

  2. Map the given point using the transformation

    (4, 2)↦(4, 4)\left(4,\ 2\right)\mapsto\left(4,\ 4\right)

    Transform the x- and y-coordinates according to the equation.

  3. State the image of the point

    (4, 4)\left(4,\ 4\right)

    This is the required image point.

Answer
(4, 4)\left(4,\ 4\right)
Question 4
2 markseasy
The point (6, −3)\left(6,\ -3\right) lies on the curve y=f(x)y=f(x). Find the coordinates of its image under the transformation y=f(2x)y=f{\left(2 x \right)}.

Worked solution

  1. Write down the transformation applied to y=f(x)

    y=f(2x)y=f{\left(2 x \right)}

    This equation defines the transformed curve.

  2. Map the given point using the transformation

    (6, −3)↦(3, −3)\left(6,\ -3\right)\mapsto\left(3,\ -3\right)

    Transform the x- and y-coordinates according to the equation.

  3. State the image of the point

    (3, −3)\left(3,\ -3\right)

    This is the required image point.

Answer
(3, −3)\left(3,\ -3\right)
Question 5
2 markseasy
The point (−2, 4)\left(-2,\ 4\right) lies on the curve y=f(x)y=f(x). Find the coordinates of its image under the transformation y=−f(x)y=- f{\left(x \right)}.

Worked solution

  1. Write down the transformation applied to y=f(x)

    y=−f(x)y=- f{\left(x \right)}

    This equation defines the transformed curve.

  2. Map the given point using the transformation

    (−2, 4)↦(−2, −4)\left(-2,\ 4\right)\mapsto\left(-2,\ -4\right)

    Transform the x- and y-coordinates according to the equation.

  3. State the image of the point

    (−2, −4)\left(-2,\ -4\right)

    This is the required image point.

Answer
(−2, −4)\left(-2,\ -4\right)

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