Hard A-Level Further transformations Questions

Challenging, exam-style A-Level Further transformations questions with worked solutions. Stretch yourself on the hardest transformations, image-of-point, range, equation problems.

transformationsimage-of-pointrangeequationasymptotecombined
A-Level34 questionsStep-by-step solutions
Question 1
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?
Show worked solution

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
Question 2
8 markschallenging
Given that f(x)=x2f(x)=x^{2}, the transformed function is g(x)=2f(x−3)+1g(x)=2 f{\left(x - 3 \right)} + 1. 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−3)+1g(x)=2 f{\left(x - 3 \right)} + 1

    Apply the transformation to f.

  3. Substitute the rule for f

    g(x)=2(x−3)2+1g(x)=2 \left(x - 3\right)^{2} + 1

    Replace f and its argument.

  4. Expand

    g(x)=2x2−12x+19g(x)=2 x^{2} - 12 x + 19

    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=1d=1

    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=−3c=-3

    c shifts the input before f is applied.

  10. 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.

  11. Describe the effect on a general x-coordinate

    x↦x+3x\mapsto x + 3

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

  12. State the vertical scale factor

    ∣a∣=2\left|a\right|=2

    The graph is stretched vertically by this factor.

  13. State the horizontal scale factor

    ∣1b∣=1\left|\tfrac{1}{b}\right|=1

    The graph is stretched horizontally by this factor.

  14. Check for reflections

    no reflection\text{no reflection}

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

  15. Select the correct expression

    g(x)=2x2−12x+19g(x)=2 x^{2} - 12 x + 19

    This matches the transformed function.

Answer
2x2−12x+192 x^{2} - 12 x + 19
Question 3
8 markschallenging
Describe fully the sequence of transformations that maps y=f(x)y=f(x) onto y=3f(x)+4y=3 f{\left(x \right)} + 4.
Show worked solution

Worked solution

  1. Compare the equations

    y=f(x) → y=3f(x)+4y=f(x)\ \to\ y=3 f{\left(x \right)} + 4

    Identify every change between the two equations.

  2. Separate the vertical scaling and translation

    y↦3y+4y\mapsto 3 y + 4

    The output is multiplied by 3 and then shifted by 4.

  3. 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.

  4. Identify the vertical factor a

    a=3a=3

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

  5. Identify the vertical shift d

    d=4d=4

    d translates the graph up, or down when negative.

  6. Identify the horizontal factor b

    b=1b=1

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

  7. Identify the horizontal constant c

    c=0c=0

    c shifts the input before f is applied.

  8. Describe the effect on a general y-coordinate

    y↦3y+4y\mapsto 3 y + 4

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

  9. 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.

  10. State the vertical scale factor

    ∣a∣=3\left|a\right|=3

    The graph is stretched vertically by this factor.

  11. State the horizontal scale factor

    ∣1b∣=1\left|\tfrac{1}{b}\right|=1

    The graph is stretched horizontally by this factor.

  12. Check for reflections

    no reflection\text{no reflection}

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

  13. Apply changes inside f first

    bx+c=xbx+c=x

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

  14. Write the mapping of a general point

    (x, y)↦(x, 3y+4)(x,\ y)\mapsto\left(x,\ 3 y + 4\right)

    This sends every point of y=f(x) to the transformed curve.

  15. State the transformations in order

    stretch ×3, then shift 4\text{stretch }\times 3\text{, then shift }4

    Apply the stretch before the vertical translation.

Answer
Vertical stretch scale factor 3, then translation by (0, 4)
Question 4
8 markschallenging
State the correct sequence of transformations that maps y=f(x)y=f(x) onto y=2f(x)−3y=2 f{\left(x \right)} - 3.
Show worked solution

Worked solution

  1. Write the target equation

    y=2f(x)−3y=2 f{\left(x \right)} - 3

    We must reach this from y=f(x).

  2. Note that outside changes act after f

    y↦2y−3y\mapsto 2 y - 3

    Vertical changes apply to the whole function once f has been evaluated.

  3. 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.

  4. Identify the vertical factor a

    a=2a=2

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

  5. Identify the vertical shift d

    d=−3d=-3

    d translates the graph up, or down when negative.

  6. Identify the horizontal factor b

    b=1b=1

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

  7. Identify the horizontal constant c

    c=0c=0

    c shifts the input before f is applied.

  8. Describe the effect on a general y-coordinate

    y↦2y−3y\mapsto 2 y - 3

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

  9. 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.

  10. State the vertical scale factor

    ∣a∣=2\left|a\right|=2

    The graph is stretched vertically by this factor.

  11. State the horizontal scale factor

    ∣1b∣=1\left|\tfrac{1}{b}\right|=1

    The graph is stretched horizontally by this factor.

  12. Check for reflections

    no reflection\text{no reflection}

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

  13. Apply changes inside f first

    bx+c=xbx+c=x

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

  14. Write the mapping of a general point

    (x, y)↦(x, 2y−3)(x,\ y)\mapsto\left(x,\ 2 y - 3\right)

    This sends every point of y=f(x) to the transformed curve.

  15. State the correct order

    stretch first, then translate\text{stretch first, then translate}

    The stretch must be applied before the vertical translation.

Answer
Vertical stretch scale factor 2, then translate by (0, -3)
Question 5
8 markschallenging
The function ff is defined on the domain 0≤x≤100\le x\le 10. Find the domain of y=f(2x−4)y=f{\left(2 x - 4 \right)}.
Show worked solution

Worked solution

  1. State the given domain of f

    0≤x≤100\le x\le 10

    f only accepts inputs in this interval.

  2. Identify the input to f

    2x−42 x - 4

    In the new function the input to f is bx+c.

  3. Require the input to lie in f's domain

    0≤2x−4≤100\le 2 x - 4\le 10

    The transformed function is only defined when its input stays in f's domain.

  4. Solve the inequality for x

    2≤x≤72\le x\le 7

    Rearranging isolates the allowed values of x.

  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=1a=1

    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=2b=2

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

  9. Identify the horizontal constant c

    c=−4c=-4

    c shifts the input before f is applied.

  10. Describe the effect on a general y-coordinate

    y↦yy\mapsto y

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

  11. Describe the effect on a general x-coordinate

    x↦x2+2x\mapsto \frac{x}{2} + 2

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

  12. State the vertical scale factor

    ∣a∣=1\left|a\right|=1

    The graph is stretched vertically by this factor.

  13. State the horizontal scale factor

    ∣1b∣=12\left|\tfrac{1}{b}\right|=\frac{1}{2}

    The graph is stretched horizontally by this factor.

  14. Check for reflections

    no reflection\text{no reflection}

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

  15. State the domain of the transformed function

    2≤x≤72\le x\le 7

    This is the domain of the new function.

Answer
2≤x≤72\le x\le 7

Unlock 29 more Further transformations questions

Create a free account to work through every A-Level Further transformations question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More Further transformations practice

Practice QuestionsWorked Solutions

Related Pure Maths topics

Areas and further applicationsMathematical argument and notationArithmetic sequences and seriesBinomial expansion (rational powers)Binomial expansionCircles