Graphing Functions (Edexcel IGCSE Further Maths)

Spec point — 4A

Graphing Functions

How do I graph the function y = f(x)?

• A point $(a,b)$ lies on the graph of $y=f(x)$ if $f(a)=b$

• The horizontal axis is used for the domain

  • these are the *x-*values

• The vertical axis is used for the range

  • these are the corresponding *y-*values

• Graphing a function can involve any or all of the following skills

  • creating a table of values
  • knowing the general shapes and characteristics of different types of graphs
  • finding axis intercepts
  • finding turning points (including maxima and minima)
  • identifying asymptotes

What is the difference between “draw” and “sketch”?

• 'Sketch' questions will not provide a detailed grid to draw the graph on

  • A sketch is meant to show key features of a graph

    • It does not need to be perfectly to scale
    • It does not need to plot points precisely

  • When asked to sketch you should:

    • Show the correct general shape
    • Label any key points such as the intersections with the axes
    • Label the axes
    • Label the graph by giving its function or equation

• 'Draw' questions will always provide a detailed grid to draw the graph on

  • A drawing of a graph is meant to be as accurate as possible

  • When asked to draw you should:

    • Use a ruler for any straight lines
    • Draw to scale
    • Plot any points precisely
    • Join points with a straight line or smooth curve
    • Label any key points such as the intersections with the axes
    • Label the axes
    • Label the graph by giving its function or equation

Can my calculator help me sketch/draw a graph?

• Your calculator may be able to display graphs of functions

  • If so make sure you are familiar with how this feature works

• This can be useful for checking your work

• It can also help with identifying

  • the general shape of a graph
  • axis intercepts
  • turning points (including maxima and minima)

• But be careful, especially on 'show that' questions

  • Make sure key parts of an answer are backed up by working
  • Marks may depend on this

Key Features of Graphs

Spec point — 4A

What are the key features of graphs?

You should be familiar with the following key features and know how to identify them

• Local minima and maxima

  • These are points where the graph has a minimum/maximum for a local region

    • They are the 'peaks' or 'valleys' of a graph
    • They are not necessarily the minimum or maximum for the whole graph

  • They are also called turning points

  • A graph can have multiple local minima or maxima

  • They occur where the gradient of the graph is equal to zero

    • So if you are graphing $y = \mathrm{f} ( x )$
    • Turning points will occur where $\mathrm{f}^{'} ( x ) = 0$

• Intercepts

  • y­­-intercepts are where the graph crosses the y-axis

    • At these points $x=0$

  • x-intercepts are where the graph crosses the x-axis

    • At these points y = 0
    • These points are also called the roots of the function

• Symmetry

  • Some graphs have lines of symmetry

    • E.g. a quadratic graph has a vertical line of symmetry through its minimum or maximum point

• Asymptotes

  • These are lines which the graph will get closer to but not cross

  • These can be horizontal or vertical

    • Exponential graphs have horizontal asymptotes
    • Logarithmic graphs and graphs of rational functions have vertical asymptotes

Exam Hint

• Don't forget to label the graph, the axes and key features when drawing or sketching a graph

  • Marks may depend on this

• Your calculator may be able to help you with drawing or sketching a graph

  • But don't forget to show working, especially in a 'show that' question

• Sketching a graph can often be useful even if a question doesn't require it

Worked Example A function is defined by $\mathrm{f} ( x ) = x^{2} - 4 x - 5$.

Sketch the graph $y=f(x)$.

Remember, 'sketch' here means we only need to show the general shape and key features

This is a positive quadratic, so it is going to be a 'u-shaped' parabola

*Find the y-intercept by substituting in *$x=0$

$\mathrm{f} ( 0 ) = ( 0 )^{2} - 4 ( 0 ) - 5 = - 5$

*So the y- intercept is *$( 0 , - 5 )$

*The x-intercepts will occur when *$y = \mathrm{f} ( x ) = 0$

$\begin{gathered} x^{2} - 4 x - 5 & = 0 \\ ( x - 5 ) ( x + 1 ) & = 0 \end{gathered}$

$x=5,x=-1$

*So the x-intercepts are *$( 5 , 0 )$ and $( - 1 , 0 )$

The turning point will occur when $\mathrm{f}^{'} ( x ) = 0$

$\mathrm{f}^{'} ( x ) = 2 x - 4$

$\begin{gathered} 2 x - 4 & = 0 \\ 2 x & = 4 \\ x = 2 \end{gathered}$

Substitute into $\mathrm{f} ( x )$* to find the corresponding y- value*

$\begin{gathered} \mathrm{f} ( 2 ) & = ( 2 )^{2} - 4 ( 2 ) - 5 \\ = 4 - 8 - 5 \\ = - 9 \end{gathered}$

So the turning point is $( 2 , - 9 )$

Sketch a graph showing the correct general shape and incorporating all the key points identified above Label the graph, the axes, and the key features