Types of Graphs

Why do we need to know what graphs look like?

Graphs are used in various aspects of mathematics – but in the real world they can take on specific meanings
For example a linear (straight line) graph could be the path a ship needs to sail along to get from one port to another
An exponential graph ( $y = k^{x}$ ) can be used to model population growth – for instance to monitor wildlife conservation projects

What are the shapes of graphs that we need to know?

Recalling facts alone won’t do much for boosting your IGCSE Mathematics grade!
But being familiar with the general shapes of graphs will help you quickly recognise the sort of maths you are dealing with and features of the graph a question may refer to
Below the basic form of the five types of function (other than trig graphs) you need to recognise;
- linear ( $y = \pm x$ )
- quadratic ( $y = \pm x^{2}$ )
- cubic ( $y = \pm x^{3}$ )
- reciprocal ( $y = \pm \frac{1}{x}$ )
- exponential ( $y = k^{\pm x}$ )

w2DxgV4S_edexcel-igcse-3-graphs-types-of-graphs

You may also need to recognise the shapes of the graphs $y = x^{\frac{1}{2}}$ and $y = x^{- \frac{1}{2}}$
In addition, you need to recognise the three basic trigonometric graphs- but these are dealt with in another section

Worked example

Match the graphs to the equations.

Graphs:

Equations:

(1) $y = 0.6 x + 2$ , (2) $y = 3^{x}$ , (3) $y = - 0.7 x^{3}$ , (4) $y = \frac{4}{x}$ , (5) $y = - x^{2} + 3 x + 2$

Starting with the equations,
(1) is a linear equation (y = mx + c) so matches the only straight line, graph (D)
(2) is an exponential equation with a positive coefficient so matches graph (A)
(3) is a cubic equation with a negative coefficient so matches graph (E)
(4) is a reciprocal equation (notice that it takes the same form as inverse proportion) with a positive coefficient so matches graph (B)
(5) is a quadratic equation with a negative coefficient so matches graph (C)

Graph (A) → Equation (2)

Graph (B) → Equation (4)

Graph (C) → Equation (5)

Graph (D) → Equation (1)

Graph (E) → Equation (3)

Drawing Graphs Using a Table

How do we draw a graph using a table of values without using a calculator?

Before you start, think what the graph might look like- see the previous notes on being familiar with shapes of graphs
Using the rules of BIDMAS/ order of operations, substitute each $x$ - value into the given function
PLOT POINTS and join with a SMOOTH CURVE
If there are any points that don't seem to fit with the shape of the rest of the curve, check your calculations for them again!

How do we draw a graph using a table of values with a calculator?

Before you start, think what the graph might look like – see the previous notes on being familiar with shapes of graphs
Find the TABLE function on your CALCULATOR
Enter the FUNCTION – f(x)
(use ALPHA button and x or X, depending on make/model)
(Press = when finished)
(If you are asked for another function, g(x), just press enter again)
Enter Start, End and Step (gap between x values)
Press = and scroll up and down to see y values
PLOT POINTS and join with a SMOOTH CURVE
To avoid errors always put negative numbers in brackets and use the (-) key rather than the subtraction key
If your calculator does not have a TABLE function, then you will have to work out each y value separately using the normal mode on your calculator

Exam Tip

When using the TABLE function of your calculator, double-check that your calculator's y-values are the same as any that are given in the question
Be prepared to draw graphs in either the calculator or non-calculator paper and make sure you are familiar with both methods

Worked example

Calculator Allowed

(a)

Complete the table of values for the function

y = x^{3} - 5 x + 2

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$y$		$4$					$14$

Use the TABLE function on your calculator for

f (x) = x^{2} - x - 6

, starting at -3, ending at 3 and with steps of 1

If your calculator does not have a TABLE function then substitute the values of $x$ into the function one by one for the missing values, being careful to put negative numbers in brackets, e.g.

x = - 3, y = {(- 3)}^{3} - 5 (- 3) + 2 = - 10

$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$y$	$- 10$	$4$	$6$	$2$	$- 2$	$0$	$14$

(b)

On the grid provided, draw the graph of

y = x^{3} - 5 x + 2

for values of

x

from

- 3

3

Carefully plot the points from your table of values in (a) on the grid, noting the different scales on the and axes
For example, the first column represents the point $(- 3, - 10)$

After plotting the points, join them with a smooth curve- do not use a ruler!

2-14-drawing-graphs

It is best practice to label the curve with its equation

Types of Graphs (CIE IGCSE Maths: Extended)

Revision Note

Types of Graphs

Why do we need to know what graphs look like?

What are the shapes of graphs that we need to know?

Worked example

Drawing Graphs Using a Table

How do we draw a graph using a table of values without using a calculator?

How do we draw a graph using a table of values with a calculator?

Exam Tip

Worked example

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Author: Amber