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A point can be placed on the grid using two numbers (coordinates) which describe its position relative to the x and y axes.

The coordinates of the origin are \((0, 0)\).

To plot the point \((4, 3)\), for example, start at \((0, 0)\), move 4 units to the right along the x-axis and then 3 units up.

Example

On the grid below, what are the coordinates of point A?

Solution

The coordinates of point A are \((3, 4)\) because the x-coordinate is 3 and the y-coordinate is 4.

When plotting points, it may be helpful to find the x-coordinate on the horizontal number line and then move either up for a positive y-coordinate or down for a negative y-coordinate.

Example

On the grid above, which point has the coordinates \((-5, 2)\)?

Solution

B is the point \((-5, 2)\).

Remember, the x-coordinate always comes first.

Look at the coordinates of points A, B, C and D on the grid.

For all these points

• they are on the y axis
• their x-coordinate is 0

At every point on the y-axis (not just at A, B, C and D), the x coordinate is 0.

The equation of the y axis is \(x = 0\)

Look at the coordinates of points E, F, G and H on the grid.

For all these points

• they are on the x axis
• their y-coordinate is 0

At every point on the x-axis (not just at E, F, G and H), the y-coordinate is 0.

The equation of the x axis is \(y = 0\)

Look at the coordinates of points A, B, C, D and E on the grid.

For all these points

• they all lie on the purple vertical line
• their x-coordinate is 2

At every point on the line (not just at A, B, C, D and E), the x-coordinate is 2.

The equation of the purple line is \(x= 2\)

Look at the line labelled A on the grid above.

At every point on this line, the y-coordinate is 5.

The equation of line A is \(y = 5\)

Example

What is the equation of line B?

Solution

The equation is \(y = 2\). At every point on the line, the y-coordinate is 2.

Summary

• The equation of the x-axis is \(y = 0\)
• The equation of the y-axis is \(x = 0\)
• Any line parallel to the x-axis is horizontal has an equation of the form \(y = a\)
• Any line parallel to the y-axis is vertical and has an equation of the form \(x = a\)

In general, the equation of a straight line takes the form \(y = mx + c\) where \(m\) and \(c\) are numbers which can be positive or negative.

\(m\) and \(c\) can also equal zero such as in the equations \(y = 3\) (\(m = 0\) and \(c = 3\)) or \(y = -5x\) (\(m = –5\) and \(c = 0\)).

To draw a straight line, given its equation

• Set up a table of values for x
• Calculate and fill in the equivalent values for y
• Use the values of x and y to plot at least 3 sets of coordinates
• Join the plotted points with a straight line and continue it across the grid

Example

Complete this table of values for \(y = 2x–1\). Use this to draw the graph of \(y = 2x–1\).

Solution

The table is already filled with values of \(x\), so we must calculate and fill in the equivalent values for \(y\).

• Start with the first \(x\) value.

\(x = –2\)

When \(x = –2\), \(y = 2(–2) – 1\)

\(y = –4 –1 = – 5\)

• Put \(y = –5\) into the table below \(x = – 2\)

• Do the same with the other values of \(x\).

When \(x = 0\)

\(y = 2(0) – 1\)

\(y = –1\)

When \(x = 2\)

\(y = 2(2) – 1\)

\(y = 3\)

When \(x = 4\)

\(y = 2(4) – 1\)

\(y = 7\)

Use the values of \(x\) and \(y\) to plot the coordinates. Plot the points \((−2, −5), (0, −1), (2, 3)\) and \((4, 7)\).