Algebra

Coordinates and Linear Graphs

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to plot coordinates in all four quadrants.
✓By the end of this lesson students will be able to understand the terms 'gradient' and 'y-intercept'.
✓By the end of this lesson students will be able to identify the gradient and y-intercept from an equation of the form y = mx + c.
✓By the end of this lesson students will be able to plot linear graphs from an equation of the form y = mx + c using a table of values.
✓By the end of this lesson students will be able to plot linear graphs using the gradient and y-intercept.

Key concepts

Coordinates

Coordinates are a pair of numbers (x, y) that describe the exact position of a point on a coordinate plane. The first number, x, tells you the horizontal position (left or right from the origin), and the second number, y, tells you the vertical position (up or down from the origin). The origin is the point (0, 0) where the x-axis and y-axis intersect. The plane is divided into four quadrants.

Linear Graph

A linear graph is a graph that forms a straight line when plotted on a coordinate plane. The relationship between the x and y values in a linear graph can be described by a linear equation.

Equation of a Straight Line (y = mx + c)

This is the general form of the equation for a straight line. It shows the relationship between the x and y coordinates for any point on the line.

y = mx + c

Gradient (m)

The gradient, represented by 'm' in the equation y = mx + c, measures the steepness and direction of a line. A positive gradient means the line slopes upwards from left to right, while a negative gradient means it slopes downwards. It is calculated as the 'change in y' divided by the 'change in x' between any two points on the line.

m = (change in y) / (change in x)

Y-intercept (c)

The y-intercept, represented by 'c' in the equation y = mx + c, is the point where the line crosses the y-axis. At this point, the x-coordinate is always 0. So, the y-intercept is the point (0, c).

Key facts to remember

1Coordinates are written as (x, y), where x is the horizontal position and y is the vertical position.
2The x-axis is horizontal, and the y-axis is vertical. The origin is the point (0, 0).
3A linear graph is a graph that forms a straight line.
4The general equation of a straight line is y = mx + c.
5'm' represents the gradient (steepness) of the line.
6'c' represents the y-intercept (where the line crosses the y-axis, i.e., the point (0, c)).
7Gradient = (change in y) / (change in x).
8A positive gradient means the line slopes upwards from left to right; a negative gradient means it slopes downwards.

Worked examples

Example 1

Plot the graph of y = 2x + 1 for x values from -2 to 2.

ICreate a table of values for x and y:

IIWhen x = -2, y = 2(-2) + 1 = -4 + 1 = -3. Point: (-2, -3)

IIIWhen x = -1, y = 2(-1) + 1 = -2 + 1 = -1. Point: (-1, -1)

IVWhen x = 0, y = 2(0) + 1 = 0 + 1 = 1. Point: (0, 1)

VWhen x = 1, y = 2(1) + 1 = 2 + 1 = 3. Point: (1, 3)

VIWhen x = 2, y = 2(2) + 1 = 4 + 1 = 5. Point: (2, 5)

VIIPlot these points on a coordinate grid.

VIIIDraw a straight line through all the plotted points using a ruler, extending it across the full range of the grid.

Answer

The graph is a straight line passing through (-2, -3), (-1, -1), (0, 1), (1, 3), and (2, 5).

Using a table of values helps ensure accuracy by calculating several points.

Example 2

Identify the gradient and y-intercept of the line y = -3x + 4, then use them to plot the graph.

ICompare the equation y = -3x + 4 with the general form y = mx + c.

IIIdentify the gradient (m): m = -3.

IIIIdentify the y-intercept (c): c = 4. This means the line crosses the y-axis at (0, 4).

IVPlot the y-intercept (0, 4) on the coordinate grid.

VFrom the y-intercept (0, 4), use the gradient. A gradient of -3 can be written as -3/1. This means 'rise' of -3 (go down 3 units) and 'run' of 1 (go right 1 unit).

VIFrom (0, 4), move down 3 units and right 1 unit to find the next point: (0+1, 4-3) = (1, 1).

VIIRepeat this process: from (1, 1), move down 3 units and right 1 unit to find (2, -2).

VIIITo find points to the left, reverse the gradient: 'rise' of 3 (go up 3 units) and 'run' of -1 (go left 1 unit).

9From (0, 4), move up 3 units and left 1 unit to find (-1, 7).

10Plot these points and draw a straight line through them using a ruler.

Answer

Gradient = -3, Y-intercept = 4. The graph is a straight line passing through points such as (-1, 7), (0, 4), (1, 1), and (2, -2).

The gradient can be thought of as 'rise over run'. A negative gradient means the line slopes downwards from left to right.

Example 3

Plot the graph of y = (1/2)x - 2.

IIdentify the gradient (m) and y-intercept (c) from the equation y = (1/2)x - 2.

IIGradient (m) = 1/2.

IIIY-intercept (c) = -2. This means the line crosses the y-axis at (0, -2).

IVPlot the y-intercept (0, -2) on the coordinate grid.

VFrom the y-intercept (0, -2), use the gradient 1/2. This means 'rise' of 1 (go up 1 unit) and 'run' of 2 (go right 2 units).

VIFrom (0, -2), move up 1 unit and right 2 units to find the next point: (0+2, -2+1) = (2, -1).

VIIRepeat this process: from (2, -1), move up 1 unit and right 2 units to find (4, 0).

VIIITo find points to the left, reverse the gradient: 'rise' of -1 (go down 1 unit) and 'run' of -2 (go left 2 units).

9From (0, -2), move down 1 unit and left 2 units to find (-2, -3).

10Plot these points and draw a straight line through them using a ruler.

Answer

The graph is a straight line passing through points such as (-2, -3), (0, -2), (2, -1), and (4, 0).

When the gradient is a fraction, the numerator is the 'rise' and the denominator is the 'run'.

Common mistakes

✗Swapping the x and y coordinates when plotting points (e.g., plotting (3, 2) instead of (2, 3)).
✗Incorrectly identifying the gradient (m) or y-intercept (c) from an equation, especially if the equation is not in the y = mx + c form.
✗Drawing a curve instead of a straight line, particularly when using only two points and not checking a third.
✗Not extending the line to cover the entire grid or the specified range of x-values.
✗Calculating the gradient as 'run over rise' instead of 'rise over run' (change in x / change in y).

Exam tips

★Always use a sharp pencil and a ruler to draw straight lines accurately. Inaccurate lines can lose marks.
★Label your axes clearly (x and y) and mark the scale on both axes.
★When using a table of values, show your calculations for at least a couple of points to demonstrate your method.
★Double-check your plotted points by substituting their coordinates back into the original equation to ensure they satisfy it.

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