Year 10 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 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 and solve it for y.

#### Example 11

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 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 12

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 13

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:

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 14

Sketch the graph of 7y – 5x = 35.

7y – 5x = 35

x-intercept:

y-intercept: