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Year 9 Interactive Maths - Second Edition


Sketch Graphs

Often we need to know the general shape and location of a graph.  In such cases, a sketch graph is drawn instead of plotting a number of points to obtain the graph.

Two points are needed to obtain a straight line graph.  It is simpler to find the points of intersection of the graph with the axes.  These points are called the x- and y- intercepts.

x-intercept:

The y-coordinate of any point on the x-axis is 0.  Therefore to find the x-intercept we put y = 0 in the equation of the line and solve it for x.

y-intercept:

The x-coordinate of any point on the y-axis is 0.  Therefore to find the y-intercept we put x = 0 in the equation of the line and solve it for y.

 

Example 5

Sketch the graph of y = 3x + 6.

Solution:
 y = 3x + 6

x-intercept:

 y-intercept:

 

 



Note:

We often represent the gradient and the y-intercept of the straight line by m and c respectively.


In the previous example:   

From the ongoing discussion we can infer that y = 3x + 6 is a straight line with a gradient of 3 and y-intercept of 6.


In general:

A linear function of the form

y = mx + c

represents the equation of a straight line with a gradient of m and y-intercept of c.

In the example under consideration, the gradient of the straight line is positive.  So, the straight line slopes upward as the value of x increases.

 

Example 6

Sketch the graph of y = 2x + 4.

Solution:

 y = 2x + 4

x-intercept:

 y-intercept:

 

Note:

From the ongoing discussion we find that the linear function y = 2x + 4 represents the equation of a straight line with a gradient of 2 and y-intercept of 4.

In the example under consideration, the gradient of the straight line is negative.  So, the straight line slopes downward as the value of x increases.


Example 7

Sketch the graph of y = 2x.

Solution:
 y = 2x

x-intercept:

 y-intercept:

When x = 0, y = 0.

As both the x- and y- intercepts are (0, 0), another point is needed.

We find when x = 5, y = 10.  So, (5, 10) is an example of another point that can be used to form the straight line graph.

Alternative technique:

Use the gradient-intercept method:

So, the straight line passes through (0, 0).  Use this point to draw a line of slope 2 (i.e. go across 3 units and up 6 units).

Note:

It is simpler to find the run and rise if we start from the y-intercept.


Example 8

Sketch the graph of 7y 5x = 35.

Solution:

7y 5x = 35

x-intercept:

 

 y-intercept:

 


Key Terms

sketch graph, x-intercept, y-intercept, equation of a straight line, gradient-intercept method


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