3.2 Understanding how equations are represented on a graph  (Page 2/4)

• Writing equations in the standard form: First an e x ample:

6 x + 2 y – 1 = 0 Keep the term in y on the left; move the other two to the right.

2 y = –6 x + 1 Now make the coefficient of y = 1 by dividing all the terms by 2 .

y = –3 x + ½ This the standard form.

Here m = –3 and c = ½.

• Now you practise some – also write down what m and c are, as above.

1.1 2 x + y = 3

1.2 3 y – 9 = 6 x

1.3 3 x = 6 y

1.4 2 y – 8 = 0

2 Understanding the gradient.

Previously we mentioned that the steep­ness of a graph can be calculated – this is very easy if the graph is a straight line, because it is equally steep everywhere – we say that the gradient of a straight-line graph remains constant.

Study the values of m in the six graphs in the previous exercise

If your work is correct, you will have noticed that the graphs slope up to the right where m is positive, and the graphs slope down to the right where m is negative.

In other word, m tells us about the gradient. (What do you think happens in y = 4, the odd one out?)

• With m positive, the number of units the line goes up for every unit it goes right gives us m (the gradient). When m is negative, we count how many units the line goes down for every unit it goes right.

• Here are two examples. By completing right-angled triangles in a convenient posi­tion on the lines in the graph, we can easily calculate the two gradients, as follows:

• For the top line: $m=-\frac{2}{5}$ , because the line goes down to the right, we know the gradient is negative; 2 is the height of the triangle and 5 is its length.
• For the bottom line: $m\text{=+}\frac{6}{9}=\frac{2}{3}$ , with 6 the height of the triangle, and 9 its length. We don’t write the +, and we simplify the fraction.

2.1 Now go back to the previous six graphs and do the same so that you can confirm that the m in the equation agrees with the gradient you calculate from the graph itself. Also notice how the size of m tells you how steep the graph is.

3 Finding out where the graph cuts the y –axis (called the y –intercept):

• If you study the equations of the six graphs, you will notice that the constant term ( c ) in the standard form tells us exactly where the graph cuts the y –axis!
• For e x ample, in y = 3 x –4, the y –intercept is at –4 on the y –axis.
• Confirm that this is true for all six graphs.

• Now we have a method for drawing graphs from an equation in the standard form. We don’t have to make a table – we simply use the y –intercept (given to us by c ), and the gradient (given by m ).

• On the graph paper, mark the y –intercept. Now use the gradient in the form of a fraction; if it is a whole number, then write it with 1 as a denominator. From the y –intercept, count as many units to the right as the denominator. From there count as many units as the numerator up, if m is positive, or down, if m is negative . Here are two examples:

  

(a) $y=\frac{2}{3}x-2$

The y-intercept is –2, marked on the y-axis with a circle. The gradient is $\frac{2}{3}$ , so we move from the circle three units to the right, and then 2 units up (not down – the gradient is positive). Another circle marks the spot we end up at. And now we draw the straight line through these two spots.