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Quadratic graphs

Explore quadratic expressions through roots, turning points, symmetry and values from tables or equations.

12 free questions · Notes · Worked solutions

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What you will practise

  • Recognise a quadratic expression and its graph shape.
  • Evaluate quadratics and use tables of values.
  • Find roots by factorising simple quadratics.
  • Identify the axis of symmetry and turning point for simple forms.

Revision notes

Quadratic form

A quadratic has highest power 2 and can be written ax²+bx+c. Its graph is a parabola. If a>0 it opens upwards; if a<0 it opens downwards.

Roots and intercepts

Roots are x-values where y=0, so they are x-axis crossings. The y-intercept is found by setting x=0. A factorised form such as (x−2)(x+1) makes roots visible.

Symmetry

The parabola is symmetric about a vertical line. For roots r₁ and r₂, the axis is halfway between them: x=(r₁+r₂)/2. The turning point lies on that axis.

Common checks

A quadratic can have two, one or no real x-axis crossings. Check signs when substituting negative x because (−x)² is positive.

A worked example

For y=x²−4x+3, find the roots and the axis of symmetry.

  1. Factorise: x²−4x+3=(x−1)(x−3).
  2. Set each factor to zero: x=1 or x=3.
  3. The axis is halfway between the roots: x=(1+3)/2=2.

Roots x=1 and x=3; axis x=2

Try it yourself

These are original practice questions, not official past-paper questions. Practice points count correct final answers; working is not automatically assessed.

Choose a question to begin.
Question 1 · Standard1 practice point

For y=x²+2x−8, which pair are the roots?

Need a hint?

Find two numbers whose product is −8 and sum is 2.

Worked solution
  1. Factorise x²+2x−8=(x+4)(x−2).
  2. Set each factor to zero: x=−4 or x=2.
Question 2 · Standard1 practice point

Does y=−2x²+5 open upwards or downwards?

Need a hint?

Look at the coefficient of x².

Worked solution
  1. The coefficient is −2, which is negative.
  2. A negative leading coefficient gives a downward-opening parabola.
Question 3 · Standard1 practice point

For y=(x−4)²+2, what is the turning point?

Need a hint?

In y=(x−h)²+k the turning point is (h,k).

Worked solution
  1. The expression is already in turning-point form.
  2. h=4 and k=2, giving (4,2).
Question 4 · Standard1 practice point

What is the y-intercept of y=3x²−7x+6?

Need a hint?

Set x=0.

Worked solution
  1. At x=0, y=3(0)²−7(0)+6=6.
Question 5 · Foundation1 practice point

Evaluate y=x²−3x+2 when x=4.

Need a hint?

Substitute 4 for every x.

Worked solution
  1. y=4²−3(4)+2.
  2. y=16−12+2=6.
Question 6 · Foundation1 practice point

Find the roots of x²−9=0. Give the positive root.

Need a hint?

Use the difference of two squares.

Worked solution
  1. x²−9=(x−3)(x+3).
  2. The roots are 3 and −3; the positive root is 3.
Question 7 · Standard2 practice points

For y=x²−6x+5, find the x-coordinate of the turning point.

Need a hint?

The axis lies midway between the roots.

Worked solution
  1. Factorise: (x−1)(x−5), so roots are 1 and 5.
  2. Midpoint=(1+5)/2=3.
Question 8 · Standard2 practice points

For y=x²+4x+1, calculate y when x=−2.

Need a hint?

Remember that (−2)² is positive.

Worked solution
  1. y=(−2)²+4(−2)+1.
  2. y=4−8+1=−3.
Question 9 · Standard1 practice point

A quadratic has roots x=−2 and x=5. Find the x-coordinate of its axis of symmetry.

Need a hint?

Take the midpoint of the two roots.

Worked solution
  1. Axis x=(−2+5)/2.
  2. x=3/2=1.5.
Question 10 · Standard2 practice points

Solve x²−5x+6=0. Give the smaller root.

Need a hint?

Factorise the quadratic.

Worked solution
  1. x²−5x+6=(x−2)(x−3).
  2. Roots are 2 and 3; the smaller is 2.
Question 11 · Foundation2 practice points

For y=−x²+4x+1, find y when x=3.

Need a hint?

Substitute x=3 with the negative sign in front of x².

Worked solution
  1. y=−(3²)+4(3)+1.
  2. y=−9+12+1=4.
Question 12 · Stretch3 practice points

The curve y=x²−2x−3 crosses the x-axis at two points. Find the distance between their x-coordinates.

Need a hint?

Factorise, then take the difference between the roots.

Worked solution
  1. x²−2x−3=(x−3)(x+1).
  2. Roots are 3 and −1.
  3. Their horizontal separation is 3−(−1)=4 units.

Coverage and sources

Selected IGCSE concepts, with a mix of introductory and more demanding questions. Check your own specification. AI drafted; independent teacher review is pending. Editorial details.

Cambridge IGCSE Mathematics 0580 syllabus

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