Show Working

AlgebraFoundation and Higher (some Higher only)

Quadratic and other graphs GCSE questions with answers

8 quadratic and other graphs questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

This topic covers tables of values, reading the roots and turning point of a quadratic graph, and recognising the shapes of common graphs. On Higher you also find turning points from the equation, use the equation of a circle and its tangents, recognise exponential and trig graphs, and transform graphs such as y = f(x + 2).

Quiz yourself: 8 questions

Worked example 1: complete a table of values

Here is a table of values for y = −3x − 1. Work out the missing values. Write them in order, from left to right.

x−3−2−10123y852?−4−7?
  1. When x = 0: y = −3 × 0 − 1 = −1.
  2. When x = 3: y = −3 × 3 − 1 = −10.
  3. So the missing values are −1, −10.

Answer −1, −10

01 / Quiz yourselfTry it, then tap to check

Quadratic and other graphs questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Complete a table of valuesBoth tiers

Here is a table of values for y = −x + 5. Work out the missing values. Write them in order, from left to right.

x012345y?43?10
Show the answer and working
  1. When x = 0: y = −(0) + 5 = 5.
  2. When x = 3: y = −(3) + 5 = 2.
  3. So the missing values are 5, 2.

Answer 5, 2

02 / Recognise graph shapesBoth tiers

Which graph could be the graph of y = x3? Type the letter.

ABCD
Show the answer and working
  1. y = x3 is an S-shaped curve through the origin, rising from bottom left to top right.
  2. A is a straight line going down, so it is the graph of something like y = 3 − x.
  3. C is two separate curves in opposite corners that never touch the axes, so it is the graph of something like y = 1x.
  4. D is an upside-down U (a parabola) with its highest point at (0, 4), so it is the graph of something like y = 4 − x2.
  5. So the answer is B.

Answer B

03 / Roots and turning points of quadratic graphsBoth tiers

The grid shows the graph of y = x2 − 2x − 3. Use the graph to solve x2 − 2x − 3 = 0.

xy−2−101234−5−4−3−2−1012345
Show the answer and working
  1. The solutions of x2 − 2x − 3 = 0 are where the graph crosses the x-axis (where y = 0).
  2. It crosses at x = −1 and x = 3.

Answer x = −1, x = 3

04 / Equation of a circleHigher only

A circle has equation x2 + y2 = 144.
Write down its radius.

Show the answer and working
  1. A circle with centre (0, 0) and radius r has equation x2 + y2 = r2.
  2. Here r2 = 144, so r = √144 = 12.

Answer 12

05 / Recognise exponential and trig graphsHigher only

Which graph could be the graph of y = 2x? Type the letter.

ABCD
Show the answer and working
  1. y = 2x is a curve that stays above the x-axis, crosses the y-axis at 1 and climbs faster and faster.
  2. A is two separate curves in opposite corners that never touch the axes, so it is the graph of something like y = 1x.
  3. C is an S-shaped curve through the origin, rising from bottom left to top right, so it is the graph of something like y = x3.
  4. D is an upside-down U (a parabola) with its highest point at (0, 4), so it is the graph of something like y = 4 − x2.
  5. So the answer is B.

Answer B

06 / Turning points from the equationHigher only

Find the coordinates of the turning point of the graph of y = (x + 5)(x − 3).

Show the answer and working
  1. The graph crosses the x-axis where (x + 5)(x − 3) = 0: at x = −5 and x = 3.
  2. The turning point is halfway between the roots, on the line of symmetry: x = −5 + 32 = −1.
  3. Put x = −1 into the equation: y = (−1 + 5)(−1 − 3) = −16.
  4. The turning point is (−1, −16).

Answer (−1, −16)

07 / Transformations of graphsHigher only

The point P(4, −3) lies on the graph of y = f(x).
The graph is transformed to y = f(x + 6). Write down the coordinates of the point that P moves to.

Show the answer and working
  1. y = f(x + 6) moves the graph left by 6: a translation by \vec−60. Subtract 6 from the x-coordinate.
  2. So (4, −3) moves to (−2, −3).

Answer (−2, −3)

08 / Tangent to a circleHigher only

The point P(−1, 5) lies on the circle x2 + y2 = 26.
Find the equation of the tangent to the circle at P. Give your answer in the form y = mx + c.

xyOP(−1, 5)
Not drawn accurately
Show the answer and working
  1. The radius OP goes from (0, 0) to (−1, 5), so its gradient is 5−1 = −5.
  2. A tangent is perpendicular to the radius at the point where it touches, so its gradient is −1 ÷ (−5) = 15.
  3. Put (−1, 5) into y = 15x + c: 5 = 15 × (−1) + c, so c = 265.
  4. The tangent is y = 15x + 265.

Answer y = 15x + 265

02 / What's covered8 kinds of question

Quadratic and other graphs at GCSE and where marks go

  • Complete a table of valuesBoth tiers
  • Recognise graph shapesBoth tiers
  • Roots and turning points of quadratic graphsBoth tiers
  • Equation of a circleHigher only
  • Recognise exponential and trig graphsHigher only
  • Turning points from the equationHigher only
  • Transformations of graphsHigher only
  • Tangent to a circleHigher only

Worked example 2: tangent to a circle

The point P(4, −3) lies on the circle x2 + y2 = 25.
Find the equation of the tangent to the circle at P. Give your answer in the form y = mx + c.

xyOP(4, −3)
Not drawn accurately
  1. The radius OP goes from (0, 0) to (4, −3), so its gradient is −34 = −34.
  2. A tangent is perpendicular to the radius at the point where it touches, so its gradient is −1 ÷ (−34) = 43.
  3. Put (4, −3) into y = 43x + c: −3 = 43 × 4 + c, so c = −253.
  4. The tangent is y = 43x − 253.

Answer y = 43x − 253

A common mistake

Moving a graph the wrong way. y = f(x + 2) is the graph of y = f(x) moved 2 to the left, not 2 to the right.

03 / Practise itiPhone and iPad

More quadratic and other graphs questions in Show Working

In the app, Quadratic and other graphs makes fresh questions every time, so they never run out. Tap Show me one to see a new worked example straight away, or practise ten and open the working whenever you are stuck. Forgotten a method? Type it into Look up and the answer is there.

Coming soon to the App Store

More algebra topics

SubstitutionSimplifying expressionsPowers and indicesExpanding bracketsFactorisingSolving equationsSequencesSimultaneous equationsQuadratic equationsRearranging formulaeInequalitiesStraight-line graphsReal-life graphsFunctionsIterationAll GCSE maths topics
The worked answer in Show Working: the answer, then numbered steps in plain words

Show WorkingComing soon to the App Store