Hard A-Level Transformations of graphs Questions

Challenging, exam-style A-Level Transformations of graphs questions with worked solutions. Stretch yourself on the hardest combined transformation, expanding, completing the square, turning point problems.

combined transformationexpandingcompleting the squareturning pointreflectioncubic
A-Level34 questionsStep-by-step solutions
Question 1
9 markschallenging
The function is f(x)=x3−4xf(x)=x^{3}-4x. (a) Find the roots of ff. (b) Find the equation of y=f(x−1)y=f(x-1) in expanded form. (c) State the roots of y=f(x−1)y=f(x-1).
Show worked solution

Worked solution

  1. Set f(x)=0 for the roots

    x3−4x=0x^{3}-4x=0

    For part (a), the roots are where the curve meets the x-axis.

  2. Factor out x

    x(x2−4)=0x(x^{2}-4)=0

    Take out the common factor x.

  3. Factor the difference of squares

    x(x−2)(x+2)=0x(x-2)(x+2)=0

    x^2-4 factorises using a^2-b^2=(a-b)(a+b).

  4. Read off the roots

    x=0, 2, −2x=0,\ 2,\ -2

    A product is zero when a factor is zero; this answers part (a).

  5. Set up part (b)

    y=f(x−1)=(x−1)3−4(x−1)y=f(x-1)=(x-1)^{3}-4(x-1)

    Replace every x in f with (x-1) for the right-shift by 1.

  6. Expand the cube

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

    Use the binomial expansion of (x-1)^3.

  7. Expand the linear part

    −4(x−1)=−4x+4-4(x-1)=-4x+4

    Multiply out the second term.

  8. Collect all terms

    y=x3−3x2+3x−1−4x+4y=x^{3}-3x^{2}+3x-1-4x+4

    Add the two expansions together.

  9. Simplify

    y=x3−3x2−x+3y=x^{3}-3x^{2}-x+3

    Combine 3x-4x=-x and -1+4=3.

  10. Set up part (c)

    shift each root right by 1\text{shift each root right by }1

    A translation right by 1 moves every root by +1.

  11. Shift the roots

    0→1, 2→3, −2→−10\to1,\ 2\to3,\ -2\to-1

    Add 1 to each original root.

  12. State the new roots

    x=1, 3, −1x=1,\ 3,\ -1

    So y=f(x-1) crosses the x-axis at these values.

  13. Give the required equation

    y=x3−3x2−x+3y=x^{3}-3x^{2}-x+3

    This expanded cubic is the main answer for part (b).

  14. Inside or outside?

    inside→x-direction, outside→y-direction\text{inside}\to x\text{-direction},\ \text{outside}\to y\text{-direction}

    Decide whether each change is inside f(...) or outside it. Inside changes move the graph horizontally (and act the opposite way to the sign); outside changes move it vertically.

  15. Read the sign for direction

    +⇒up/right, −⇒down/left+\Rightarrow\text{up/right},\ -\Rightarrow\text{down/left}

    Use the sign to fix the direction. Outside: plus is up, minus is down. Inside the bracket the movement is the opposite way round, which is easy to get wrong.

  16. Follow one key point

    track vertex / intercept\text{track vertex / intercept}

    Pick one important point, such as a turning point or an axis crossing, and follow it through the transformation. If that point lands correctly the whole curve is right.

Answer
y=x3−3x2−x+3y=x^{3}-3x^{2}-x+3
Question 2
8 markschallenging
The blue curve is y=x2y=x^{2}. Select the graph of y=−2(x−1)2+3y=-2(x-1)^{2}+3.
Show worked solution

Worked solution

  1. Read the vertex form

    y=−2(x−1)2+3y=-2(x-1)^{2}+3

    Vertex form shows the vertex at (1,3) with stretch and reflection.

  2. Vertex location

    (1, 3)(1,\ 3)

    The (x-1) shifts right 1 and the +3 shifts up 3.

  3. The factor 2

    vertical stretch, factor 2\text{vertical stretch, factor }2

    The magnitude 2 makes the parabola narrower.

  4. The minus sign

    reflect in x-axis\text{reflect in }x\text{-axis}

    The negative sign turns the parabola upside down, so the vertex is a maximum.

  5. Shape summary

    ∩-shape, max (1,3)\cap\text{-shape, max }(1,3)

    It opens downwards with its highest point at (1,3).

  6. Choose the sketch

    max at (1,3), narrow\text{max at }(1,3),\ \text{narrow}

    Pick the downward, narrow parabola with maximum (1,3).

  7. Is the answer reasonable?

    sense-check the size and sign\text{sense-check the size and sign}

    Step back and ask whether the size and sign of the answer make sense for the transformation described. Unreasonable values usually mean an arithmetic error.

  8. Compare with the original

    how has each feature moved?\text{how has each feature moved?}

    Compare the transformed curve feature by feature with the original: has the vertex moved, has the width changed, has it flipped? Each should match your working.

  9. Write the answer clearly

    A narrow downward parabola with maximum at (1,3)(1,3).

    Finally, write the answer out neatly so it is unambiguous. Presenting the result clearly is part of good mathematical communication.

  10. Inside or outside?

    inside→x-direction, outside→y-direction\text{inside}\to x\text{-direction},\ \text{outside}\to y\text{-direction}

    Decide whether each change is inside f(...) or outside it. Inside changes move the graph horizontally (and act the opposite way to the sign); outside changes move it vertically.

  11. Read the sign for direction

    +⇒up/right, −⇒down/left+\Rightarrow\text{up/right},\ -\Rightarrow\text{down/left}

    Use the sign to fix the direction. Outside: plus is up, minus is down. Inside the bracket the movement is the opposite way round, which is easy to get wrong.

  12. Follow one key point

    track vertex / intercept\text{track vertex / intercept}

    Pick one important point, such as a turning point or an axis crossing, and follow it through the transformation. If that point lands correctly the whole curve is right.

  13. Effect on the intercepts

    where does it cross the axes?\text{where does it cross the axes?}

    Think about where the new curve meets the axes. Translations and stretches move intercepts in predictable ways, giving a quick check on your answer.

  14. Stretch or translation?

    scale ≠ shift\text{scale}\ \ne\ \text{shift}

    Make sure you have not mixed up a stretch with a translation. A number multiplying f scales the graph; a number added to f slides it.

  15. Mind the order

    inside and outside act independently\text{inside and outside act independently}

    When several transformations are combined, a change to x and a change to y can be done in either order, but two changes to the same direction may not commute. Keep track carefully.

  16. Picture the graph

    original (blue) vs image (pink)\text{original (blue) vs image (pink)}

    Sketching the original curve and its image on the same axes turns the algebra into a picture, which makes the shift, stretch or reflection obvious.

Answer
A narrow downward parabola with maximum at (1,3)(1,3).
Question 3
8 markschallenging
Prove that translating y=f(x)y=f(x) by (a0)\binom{a}{0} and then reflecting in the yy-axis gives the same curve as first reflecting y=f(x)y=f(x) in the yy-axis and then translating by (−a0)\binom{-a}{0}. Which working is correct?
Show worked solution

Worked solution

  1. Route 1: translate first

    y=f(x−a)y=f(x-a)

    A translation by (a,0) replaces x with (x-a).

  2. Route 1: then reflect in y-axis

    x→−x: y=f(−x−a)x\to -x:\ y=f(-x-a)

    Replace x with -x in f(x-a), giving f(-x-a).

  3. Route 2: reflect first

    y=f(−x)y=f(-x)

    A reflection in the y-axis replaces x with -x.

  4. Route 2: then translate by (-a,0)

    x→x+a: y=f(−(x+a))x\to x+a:\ y=f(-(x+a))

    A translation by (-a,0) replaces x with (x+a).

  5. Simplify Route 2

    y=f(−x−a)y=f(-x-a)

    Expanding -(x+a) gives -x-a.

  6. Compare the two routes

    f(−x−a)=f(−x−a)f(-x-a)=f(-x-a)

    Both routes give exactly the same function.

  7. Conclude

    the curves are identical\text{the curves are identical}

    Hence the two sequences of transformations produce the same graph, as required.

  8. Link to earlier work

    completing the square / plotting\text{completing the square / plotting}

    This uses skills from earlier topics such as plotting graphs and completing the square, so the same coordinate and algebra methods apply here.

  9. Avoid the classic slip

    change the correct coordinate only\text{change the correct coordinate only}

    A very common error is to change the wrong coordinate or reflect in the wrong axis. Re-read which variable the transformation acts on before writing the answer.

  10. Name the transformation

    translation / stretch / reflection\text{translation / stretch / reflection}

    State clearly what type of transformation it is, because exam marks are often awarded for the correct name and full description, not just the final numbers.

  11. Is the answer reasonable?

    sense-check the size and sign\text{sense-check the size and sign}

    Step back and ask whether the size and sign of the answer make sense for the transformation described. Unreasonable values usually mean an arithmetic error.

  12. Compare with the original

    how has each feature moved?\text{how has each feature moved?}

    Compare the transformed curve feature by feature with the original: has the vertex moved, has the width changed, has it flipped? Each should match your working.

  13. Write the answer clearly

    Both routes give y=f(−x−a)y=f(-x-a), so the curves are identical.

    Finally, write the answer out neatly so it is unambiguous. Presenting the result clearly is part of good mathematical communication.

  14. Inside or outside?

    inside→x-direction, outside→y-direction\text{inside}\to x\text{-direction},\ \text{outside}\to y\text{-direction}

    Decide whether each change is inside f(...) or outside it. Inside changes move the graph horizontally (and act the opposite way to the sign); outside changes move it vertically.

  15. Read the sign for direction

    +⇒up/right, −⇒down/left+\Rightarrow\text{up/right},\ -\Rightarrow\text{down/left}

    Use the sign to fix the direction. Outside: plus is up, minus is down. Inside the bracket the movement is the opposite way round, which is easy to get wrong.

  16. Follow one key point

    track vertex / intercept\text{track vertex / intercept}

    Pick one important point, such as a turning point or an axis crossing, and follow it through the transformation. If that point lands correctly the whole curve is right.

Answer
Both routes give y=f(−x−a)y=f(-x-a), so the curves are identical.
Question 4
8 markschallenging
The function ff has domain 0≤x≤80\le x\le 8 and range 1≤f(x)≤91\le f(x)\le 9. State the domain and range of y=f(2x)+3y=f(2x)+3.
Show worked solution

Worked solution

  1. Identify the inside change

    f(2x)f(2x)

    The 2x is a horizontal stretch factor 1/2, so it affects the domain (x-values).

  2. Transform the domain

    0≤2x≤80\le 2x\le 8

    The new function accepts x where 2x lies in the original domain.

  3. Solve for x

    0≤x≤40\le x\le 4

    Divide the inequality by 2, so the domain is halved.

  4. Identify the outside change

    +3+3

    The +3 adds to the output, so it affects the range (y-values).

  5. Transform the range

    1+3≤y≤9+31+3\le y\le 9+3

    Add 3 to both ends of the original range.

  6. Simplify the range

    4≤y≤124\le y\le 12

    So every output is lifted by 3.

  7. State both

    0≤x≤4, 4≤y≤120\le x\le4,\ 4\le y\le12

    The stretch changes the domain; the vertical shift changes the range.

  8. Mind the order

    inside and outside act independently\text{inside and outside act independently}

    When several transformations are combined, a change to x and a change to y can be done in either order, but two changes to the same direction may not commute. Keep track carefully.

  9. Picture the graph

    original (blue) vs image (pink)\text{original (blue) vs image (pink)}

    Sketching the original curve and its image on the same axes turns the algebra into a picture, which makes the shift, stretch or reflection obvious.

  10. Check by substituting

    put a value back in\text{put a value back in}

    Substitute a simple value or a known point back into the transformed equation to confirm it works. A quick numerical check catches slips.

  11. Link to earlier work

    completing the square / plotting\text{completing the square / plotting}

    This uses skills from earlier topics such as plotting graphs and completing the square, so the same coordinate and algebra methods apply here.

  12. Avoid the classic slip

    change the correct coordinate only\text{change the correct coordinate only}

    A very common error is to change the wrong coordinate or reflect in the wrong axis. Re-read which variable the transformation acts on before writing the answer.

  13. Name the transformation

    translation / stretch / reflection\text{translation / stretch / reflection}

    State clearly what type of transformation it is, because exam marks are often awarded for the correct name and full description, not just the final numbers.

  14. Is the answer reasonable?

    sense-check the size and sign\text{sense-check the size and sign}

    Step back and ask whether the size and sign of the answer make sense for the transformation described. Unreasonable values usually mean an arithmetic error.

  15. Compare with the original

    how has each feature moved?\text{how has each feature moved?}

    Compare the transformed curve feature by feature with the original: has the vertex moved, has the width changed, has it flipped? Each should match your working.

  16. Write the answer clearly

    Domain 0≤x≤40\le x\le 4, range 4≤y≤124\le y\le 12.

    Finally, write the answer out neatly so it is unambiguous. Presenting the result clearly is part of good mathematical communication.

Answer
Domain 0≤x≤40\le x\le 4, range 4≤y≤124\le y\le 12.
Question 5
8 markschallenging
The curve y=x2−10x+27y=x^{2}-10x+27 is a translation of y=x2y=x^{2}. Find the coordinates of the point where y=x2−10x+27y=x^{2}-10x+27 meets y=x2y=x^{2}.
Show worked solution

Worked solution

  1. Complete the square

    x2−10x+27=(x−5)2−25+27x^{2}-10x+27=(x-5)^{2}-25+27

    Halve -10 to -5 and subtract 25, keeping +27.

  2. Simplify

    (x−5)2+2(x-5)^{2}+2

    Because -25+27=2, the vertex is at (5,2).

  3. State the translation

    (52)\binom{5}{2}

    So y=x^2 has been moved 5 right and 2 up.

  4. Set the two curves equal

    x2−10x+27=x2x^{2}-10x+27=x^{2}

    They intersect where their y-values match.

  5. Cancel x^2

    −10x+27=0-10x+27=0

    Subtract x^2 from both sides, leaving a linear equation.

  6. Rearrange

    10x=2710x=27

    Add 10x to both sides.

  7. Solve for x

    x=2.7x=2.7

    Divide both sides by 10.

  8. Find y

    y=(2.7)2=7.29y=(2.7)^{2}=7.29

    Substitute x=2.7 into y=x^2 (the simpler curve).

  9. State the point

    (2.7, 7.29)(2.7,\ 7.29)

    So the two parabolas cross at this single point.

  10. Follow one key point

    track vertex / intercept\text{track vertex / intercept}

    Pick one important point, such as a turning point or an axis crossing, and follow it through the transformation. If that point lands correctly the whole curve is right.

  11. Effect on the intercepts

    where does it cross the axes?\text{where does it cross the axes?}

    Think about where the new curve meets the axes. Translations and stretches move intercepts in predictable ways, giving a quick check on your answer.

  12. Stretch or translation?

    scale ≠ shift\text{scale}\ \ne\ \text{shift}

    Make sure you have not mixed up a stretch with a translation. A number multiplying f scales the graph; a number added to f slides it.

  13. Mind the order

    inside and outside act independently\text{inside and outside act independently}

    When several transformations are combined, a change to x and a change to y can be done in either order, but two changes to the same direction may not commute. Keep track carefully.

  14. Picture the graph

    original (blue) vs image (pink)\text{original (blue) vs image (pink)}

    Sketching the original curve and its image on the same axes turns the algebra into a picture, which makes the shift, stretch or reflection obvious.

  15. Check by substituting

    put a value back in\text{put a value back in}

    Substitute a simple value or a known point back into the transformed equation to confirm it works. A quick numerical check catches slips.

  16. Link to earlier work

    completing the square / plotting\text{completing the square / plotting}

    This uses skills from earlier topics such as plotting graphs and completing the square, so the same coordinate and algebra methods apply here.

Answer
(2.7, 7.29)(2.7,\ 7.29)

Unlock 29 more Transformations of graphs questions

Create a free account to work through every A-Level Transformations of graphs 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 Transformations of graphs practice

Practice QuestionsWorked Solutions

Related Pure Maths topics

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