3.2 Understanding how equations are represented on a graph

Mathematics

Grade 9

Number patterns, graphs, equasions,

Statistics and probability

Module 14

Understanding how equations are represented on a graph

ACTIVITY 1

To understand how equations can be represented on a graph

[LO 2.2, 2.5, 2.6]

E x ample : The equation y = 3 x + 2 tells one how the values of x and y are connected – it shows the relationship between two variables, x and y .

• For example, if x is 5, then y can be calculated from 3 × 5 + 2, giving 17. So, we substitute the value 5 for the x , and complete the calculation.

  

• The table shows some of the answers.

The point of making a table is that it gives us coordinates, which we can plot on a set of axes. From these we can draw a graph, which is a picture of the relationship between x and y.

1 In a group of 4 or 5 learners, complete the tables from the equations below. Each one should do a different table, and then discuss the answers and copy the others’ tables into your book. Below each table is a set of axes on which to draw the graph. All of these graphs will be straight lines, so you may connect the points you plot from each table.

2 In your group, discuss what you see in your graphs. Compare the graph with the equation. Here are a few suggestions for you to investigate:

2.1 What does the coefficient of x in the equation do to the graph? What happens when the coefficient is negative?

2.2 In 1.6 the table has only two columns. Is it necessary to have more than two columns if you know that the graph will be a straight line?

2.3 Compare 1.1 and 1.5 to see if you can find out what the constant does to the graph.

• The table summarises how equations, tables and graphs agree with one another. You have to memorise this information – you can’t work correctly with graphs if you don’t know the words

ACTIVITY 2

To understand and apply all the characteristics of the straight-line graph

[LO 2.5]

1 In the previous activity you used equations to make tables from which graphs were drawn. One can also draw a graph from an equation without making a table. As we saw in graph 1.6 above, if we know that we have to draw a straight line, then two points on the graph paper are sufficient to enable us to draw the line. We don’t need a table if we have the equation of the straight line.

• In this section we will refer repeatedly to the six graphs in the previous e x ercise.

• First we have to examine the structure of the equation:

y = mx + c is the standard form of the equation:

• y is on the left of the equal sign, and y has no coefficient (which is the same as saying that its coefficient is 1 , which we don’t write)
• After the equal sign there may be one or two terms (compare equations 1.5 and 1.6 above). If there are two terms, then the term in x is written first – the x may have any number as a coefficient: positive, negative or fractional.
• In the standard form we write an m to stand for the coefficient. If the coefficient is 1 , then again we don’t write it.
• The c stands for a constant – any number that is not a variable: it can also be negative, positive or fractional.
• If we write y = mx + c then this is the general equation for all straight lines. But y = –3 x + 2 is the defining equation for a specific straight line.