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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.

The graph shows x = 0 on the y-axis and y = 0 on the x-axis.

 

Example 11

Sketch the graph of y = 3x + 6.

Solution:
 y = 3x + 6

x-intercept:

When y = 0, x = -2

 y-intercept:

When x = 0, y = 6

 

The sketch graph on the Cartesian plane that goes through (-2, 0) and (0, 6).


Note:

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


In the previous example:   

m = Rise / Run = (6 - 0) / (0 - (-2)) = 6 / 2 = 3

And c = y-intercept = 6

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:

When y = 0, x = 2

 y-intercept:

When x = 0, y = 4

The sketch graph of the line that passes through the points (2, 0) and (0, 4).

 

Note:

c = y-intercept = 4

And m = rise/run = (0 - 4) / (2 - 0) = -4 / 2 = -2

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:

When y = 0, x = 0

 y-intercept:

When x = 0, y = 0.

The graph of y = 2x passes through the origin at (0, 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:

Comparing y = 2x with y = mx + c gives m = 2, c = 0.

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).

The graph of y = 2x that highlights a rise of 6 for every run of 3.

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.

Solution:

7y – 5x = 35

x-intercept:

When y = 0, x = -7

 y-intercept:

When x = 0, y = 5

The sketch graph of 7y - 5x = 35 that has an x-intercept of -7 and a y-intercept of 5.


Key Terms

sketch graph, x-intercept,  y-intercept, gradient-intercept method


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