A-Level Further transformations Practice Questions

Free A-Level Further transformations practice questions with full step-by-step worked solutions. Covers transformations, image-of-point, describe, range. Practise exam-style problems and check your method.

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.
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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
Given that f(x)=x3f(x)=x^{3}, the transformed function is g(x)=f(x+1)g(x)=f{\left(x + 1 \right)}. Which of the following is the correct expression for g(x)g(x)?
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Worked solution

  1. Write down f(x)

    f(x)=x3f(x)=x^{3}

    This is the original function.

  2. Write g(x) symbolically

    g(x)=f(x+1)g(x)=f{\left(x + 1 \right)}

    Apply the transformation to f.

  3. Select the correct expression

    g(x)=x3+3x2+3x+1g(x)=x^{3} + 3 x^{2} + 3 x + 1

    This matches the transformed function.

Answer
x3+3x2+3x+1x^{3} + 3 x^{2} + 3 x + 1
Question 3
3 marksintermediate
Given that f(x)=x2f(x)=x^{2}, the transformed function is g(x)=f(2x)g(x)=f{\left(2 x \right)}. Which of the following is the correct expression for g(x)g(x)?
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Worked solution

  1. Write down f(x)

    f(x)=x2f(x)=x^{2}

    This is the original function.

  2. Write g(x) symbolically

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

    Apply the transformation to f.

  3. Substitute the rule for f

    g(x)=4x2g(x)=4 x^{2}

    Replace f and its argument.

  4. Recall the general combined transformation

    y=a f(bx+c)+dy=a\,f(bx+c)+d

    Any combination of stretches, reflections and translations of y=f(x) fits this template.

  5. Identify the vertical factor a

    a=1a=1

    a multiplies every output, giving a vertical stretch (and a reflection when negative).

  6. Select the correct expression

    g(x)=4x2g(x)=4 x^{2}

    This matches the transformed function.

Answer
4x24 x^{2}
Question 4
5 markshard
Given that f(x)=x2f(x)=x^{2}, the transformed function is g(x)=2f(x−1)g(x)=2 f{\left(x - 1 \right)}. Which of the following is the correct expression for g(x)g(x)?
Show worked solution

Worked solution

  1. Write down f(x)

    f(x)=x2f(x)=x^{2}

    This is the original function.

  2. Write g(x) symbolically

    g(x)=2f(x−1)g(x)=2 f{\left(x - 1 \right)}

    Apply the transformation to f.

  3. Substitute the rule for f

    g(x)=2(x−1)2g(x)=2 \left(x - 1\right)^{2}

    Replace f and its argument.

  4. Expand

    g(x)=2x2−4x+2g(x)=2 x^{2} - 4 x + 2

    Simplify to a single expression.

  5. Recall the general combined transformation

    y=a f(bx+c)+dy=a\,f(bx+c)+d

    Any combination of stretches, reflections and translations of y=f(x) fits this template.

  6. Identify the vertical factor a

    a=2a=2

    a multiplies every output, giving a vertical stretch (and a reflection when negative).

  7. Identify the vertical shift d

    d=0d=0

    d translates the graph up, or down when negative.

  8. Identify the horizontal factor b

    b=1b=1

    b acts inside f, controlling the horizontal stretch and any reflection in the y-axis.

  9. Identify the horizontal constant c

    c=−1c=-1

    c shifts the input before f is applied.

  10. Select the correct expression

    g(x)=2x2−4x+2g(x)=2 x^{2} - 4 x + 2

    This matches the transformed function.

Answer
2x2−4x+22 x^{2} - 4 x + 2
Question 5
8 markschallenging
The function ff has range −2≤f(x)≤4-2\le f(x)\le 4. The function gg is defined by g(x)=2f(x)−1g(x)=2 f{\left(x \right)} - 1. Which statement about the range of gg is correct?
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Worked solution

  1. State the range of f

    −2≤f(x)≤4-2\le f(x)\le 4

    These are the outputs of f.

  2. Apply the transformation to the outputs

    y↦2y−1y\mapsto 2 y - 1

    Scale by 2 and shift by -1.

  3. Transform the boundary values

    −2↦−5,4↦7-2\mapsto -5,\quad 4\mapsto 7

    Apply the mapping to each end of the range.

  4. Recall the general combined transformation

    y=a f(bx+c)+dy=a\,f(bx+c)+d

    Any combination of stretches, reflections and translations of y=f(x) fits this template.

  5. Identify the vertical factor a

    a=2a=2

    a multiplies every output, giving a vertical stretch (and a reflection when negative).

  6. Identify the vertical shift d

    d=−1d=-1

    d translates the graph up, or down when negative.

  7. Identify the horizontal factor b

    b=1b=1

    b acts inside f, controlling the horizontal stretch and any reflection in the y-axis.

  8. Identify the horizontal constant c

    c=0c=0

    c shifts the input before f is applied.

  9. Describe the effect on a general y-coordinate

    y↦2y−1y\mapsto 2 y - 1

    Every output is scaled by a and then shifted by d.

  10. Describe the effect on a general x-coordinate

    x↦xx\mapsto x

    Solving bx+c for the original input shows where each x-coordinate moves.

  11. State the vertical scale factor

    ∣a∣=2\left|a\right|=2

    The graph is stretched vertically by this factor.

  12. State the horizontal scale factor

    ∣1b∣=1\left|\tfrac{1}{b}\right|=1

    The graph is stretched horizontally by this factor.

  13. Check for reflections

    no reflection\text{no reflection}

    A negative scale factor flips the graph in the corresponding axis.

  14. Apply changes inside f first

    bx+c=xbx+c=x

    Horizontal changes inside the bracket act before the vertical changes outside it.

  15. State the new range

    −5≤y≤7-5\le y\le 7

    This determines the correct comparison statement.

Answer
The range of g is -5 ≤ y ≤ 7

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