Quadratic & Polynomial Graphs (AQA GCSE Further Maths)

Quadratic Graphs

A quadratic is a function of the form  where  is not zero
They are a very common type of function in mathematics, so it is important to know their key features

What does a quadratic graph look like?

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

• The parabola shape of a quadratic graph can either look like a “u-shape” or an “n-shape”
  

  • A quadratic with a positive coefficient of  will be a u-shape

  • A quadratic with a negative coefficient of  will be an n-shape

• A quadratic will always cross the -axis

• A quadratic may cross the -axis twice, once, or not at all

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

• If the quadratic is a u-shape, it has a minimum point (the bottom of the u)

• If the quadratic is an n-shape, it has a maximum point (the top of the n)

• Minimum and maximum points are both examples of turning points

How do I sketch a quadratic graph?

• We could create a table of values for the function and then plot it accurately, however we often only require a sketch to be drawn, showing just the key features

• The most important features of a quadratic are

  • Its overall shape; a u-shape or an n-shape

  • Its -intercept

  • Its -intercept(s), these are also known as the roots

  • Its minimum or maximum point (turning point)

• If it is a positive quadratic ( in  is positive) it will be a u-shape

• If it is a negative quadratic ( in is negative) it will be an n-shape

• The -intercept of will be 

• The roots, or the -intercepts will be the solutions to ; 

  • You can solve a quadratic by factorising, completing the square, or using the quadratic formula

  • There may be 2, 1, or 0 solutions and therefore 2, 1, or 0 roots

• The minimum or maximum point of a quadratic can be found by;

  • Completing the square

    • Once the quadratic has been written in the form , the minimum or maximum point is given by 

    • Be careful with the sign of the x-coordinate. E.g. if the equation is then the minimum point is  but if the equation is  then the minimum point is 

  • Using differentiation

    • Solving  will find the -coordinate of the minimum or maximum point

    • You can then substitute this into the equation of the quadratic to find the -coordinate

Polynomial Graphs

What is the graph of a polynomial?

• Remember a polynomial is any finite function with non-negative indices, that could mean a quadratic, cubic, quartic or higher power

• When asked to sketch a polynomial you'll need to think about the following

  • y-axis intercept

  • x-axis intercepts (roots)

  • turning points (maximum and/or minimum)

  • a smooth curve (this takes practice!)

How do I sketch a graph of a polynomial?

STEP 1        Find the y-axis intercept by setting x = 0

STEP 2        Find the x-axis intercepts (roots) by setting y = 0

STEP 3        Consider the shape and “start”/”end” of the graph

eg. a positive cubic graph starts in third quadrant (bottom left) and “ends” in first quadrant (top right)
Whereas a negative cubic graph starts in the second quadrant (top left) and ends in the fourth quadrant (bottom right)

STEP 4        Consider where any turning points should go

STEP 5        Draw with a smooth curve

How do I sketch a polynomial from its turning points?

• Turning points are where the gradient of the graph is equal to zero

  • They are either a minimum; a u-shape

  • Or a maximum; an n-shape

• The turning points of a polynomial may be found by using differentiation, or you may be told them in the question

• First consider the order (power) of the polynomial you are sketching

• This will tell you the general shape of the graph, and how many turning points there are likely to be 

  • A quadratic will have 1 turning point

    • A positive quadratic will have a minimum

    • A negative quadratic will have a maximum

  • A cubic will have either 2 or 0 turning points

  • A quartic will have either 3 or 1 turning points

• Knowing the type of turning point is also useful

  • For a maximum

    • The graph will be increasing (going upwards from left to right) before the maximum

    • The graph will be decreasing (going downwards from left to right) after the maximum

  • For a minimum

    • The graph will be decreasing before the minimum

    • The graph will be increasing after the minimum

  • Putting a point on your sketch where each turning point is, and sketching a small u for a minimum, or a small n for a maximum will help to then fill in the overall shape of the graph