Quadratic Graphs (Cambridge (CIE) O Level Maths): Revision Note

Quadratic Graphs

What is a quadratic graph?

• A quadratic graph has the form  

  • where  is not zero

What does a quadratic graph look like?

• A quadratic graph is a smooth curve with a vertical line of symmetry

  • A positive number in front of  gives a u-shaped curve

  • A negative number in front of  gives an n-shaped curve

• The shape made by a quadratic graph is known as a parabola

• A quadratic graph will always cross the -axis

• A quadratic graph crosses the -axis twice, once, or not at all

  • The points where the graph crosses the -axis are called the roots

• If the graph is a u-shape, it has a minimum point

• If the graph is an n-shape, it has a maximum point

• Minimum and maximum points are both examples of turning points

  • A turning point can also be called a vertex

How do I sketch a quadratic graph?

• It is important to know how to sketch a quadratic curve

  • A simple drawing showing the key features is often sufficient

  • (For a more accurate graph, create a table of values and plot the points)

• To sketch a quadratic graph:

  • First sketch the and -axes

  • Identify the -intercept and mark it on the -axis

    • The -intercept of will be 

    • It can also be found by substituting in

  • Find all root(s) (0, 1 or 2) of the equation and mark them on the -axis

    • The roots will be the solutions to ; 

    • You can find the solutions by factorising, completing the square or using the quadratic formula

  • Identify if the number in is positive or negative

    • A positive value will result in a u-shape

    • A negative value will result in an n-shape

  • Sketch a smooth curve through the and -intercepts

    • Mark on any axes intercepts

    • Mark on the coordinates of the maximum/minimum point if you know it

How do I find the coordinates of the turning point by completing the square?

• The coordinates of the turning point (vertex) of a quadratic graph can be found by completing the square

• For a quadratic graph written in the form

  • the minimum or maximum point has coordinates 

• Beware: there is a sign change for the -coordinate

  • A curve with equation , has a minimum point at

  • A curve with equation , has a minimum point at

• The value of does not affect the coordinates of the turning point but it will change the shape of the graph

  • If it is positive, the graph will be a u-shape

    • The curve has a minimum point at

  • If it is negative, the graph will be an n-shape

    • The curve has a maximum point at

How do I find the coordinates of the turning point using differentiation?

• The coordinates of the turning point (maximum/minimum) of a quadratic can be found through differentiation

• To find the coordinates of the turning point

  • Differentiate the quadratic equation

    • This will give you

  • Set   and solve for

    • The solution will be the -coordinate of the turning point

  • Substitute the value into

    • This will give you the -coordinate of the turning point

How do I find the equation of a quadratic from its graph?

• If the vertex and one other point are known

  • Use the form to fill in and

    • The vertex is at

  • Then substitute in the other known point to find

• If the roots (-intercepts) and one other point are known

  • Use the form to fill in and

    • The roots are at and

  • Then substitute in the other known point to find

• If then you only need either the vertex or the roots