Drawing Linear Graphs

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What is a Linear Graph?

A linear graph is a straight line on a graph. The word "linear" comes from "line". When you draw a linear graph, you will always get a perfectly straight line, not a curve.

A linear equation is an equation that makes a straight line when you draw it. Linear equations have variables (like x and y) but these variables are never multiplied together, squared, or cubed. They are always to the power of 1.

Examples of linear equations:

• y = 2x + 3
• y = –4x + 1
• y = 5 – 3x
• 2x + y = 7

Not linear equations (these would make curves):

• y = x² + 2 (because of the x²)
• y = 1/x (because this makes a curve)

Understanding y = mx + c

Most linear equations can be written in the form y = mx + c. This is a very useful format because it tells you two important things about the line straight away.

What do the letters mean?

• y and x are variables (they can have different values)
• m is the gradient (how steep the line is)
• c is the y-intercept (where the line crosses the y-axis)

Example: In the equation y = 3x + 2:

• The gradient (m) is 3
• The y-intercept (c) is 2
• This means the line crosses the y-axis at the point (0, 2)

Example: In the equation y = –2x + 5:

• The gradient (m) is –2
• The y-intercept (c) is 5
• This means the line crosses the y-axis at the point (0, 5)

How to Draw a Graph from y = mx + c

When you have an equation already in the form y = mx + c, follow these steps:

Method 1: Using a Table of Values

Step 1: Choose some values for x (usually start with simple numbers like –2, –1, 0, 1, 2, 3)

Step 2: Work out the matching y values by substituting each x value into the equation

Step 3: Write your results in a table

Step 4: Plot each (x, y) pair as a point on graph paper

Step 5: Draw a straight line through all the points using a ruler

Step 6: Extend the line a little bit beyond your points and add arrows at both ends to show it continues

Worked Example:

Draw the graph of y = 2x + 1

Step 1: Choose x values. Let's use x = –1, 0, 1, 2, 3

Step 2 & 3: Calculate y values and make a table:

How we got these y values:

• When x = –1: y = 2(–1) + 1 = –2 + 1 = –1
• When x = 0: y = 2(0) + 1 = 0 + 1 = 1
• When x = 1: y = 2(1) + 1 = 2 + 1 = 3
• When x = 2: y = 2(2) + 1 = 4 + 1 = 5
• When x = 3: y = 2(3) + 1 = 6 + 1 = 7

Step 4: Plot the points: (–1, –1), (0, 1), (1, 3), (2, 5), (3, 7)

Step 5 & 6: Use a ruler to draw a straight line through all points, extending it with arrows at both ends.

Method 2: Quick Method Using m and c

If you understand gradient and y-intercept well, you can draw the line more quickly:

Step 1: Plot the y-intercept (0, c) first

Step 2: Use the gradient to find another point

Step 3: Draw the line through these points

Worked Example:

Draw the graph of y = 3x – 2

Step 1: The y-intercept is –2, so plot the point (0, –2)

Step 2: The gradient is 3, which means "go up 3 and across 1 to the right"

• From (0, –2), move right 1 unit and up 3 units
• This gives you the point (1, 1)

Step 3: Draw a straight line through (0, –2) and (1, 1), extending with arrows

Drawing Graphs with Negative Gradients

When the gradient (m) is negative, the line slopes downwards from left to right.

Worked Example:

Draw the graph of y = –2x + 5

Using a table of values:

Calculations:

• When x = 0: y = –2(0) + 5 = 5
• When x = 1: y = –2(1) + 5 = –2 + 5 = 3
• When x = 2: y = –2(2) + 5 = –4 + 5 = 1
• When x = 3: y = –2(3) + 5 = –6 + 5 = –1
• When x = 4: y = –2(4) + 5 = –8 + 5 = –3

Plot these points and draw a straight line through them. Notice how the line goes down as you move from left to right.

Equations Written in Different Forms

Sometimes the equation is not already in the form y = mx + c. You might see equations like:

• y = 7 – 4x
• 3x + 2y = 5
• 2y = 6x + 8

For equations like y = 7 – 4x:

This is already nearly in the form y = mx + c. Remember that y = 7 – 4x is the same as y = –4x + 7.

So: m = –4 and c = 7

You can now draw this graph using a table of values or the quick method.

For equations like 3x + 2y = 5:

You need to rearrange this to get y by itself on one side.

Step 1: Get the y term by itself (move everything else to the other side)

• 3x + 2y = 5
• 2y = 5 – 3x (subtract 3x from both sides)

Step 2: Divide everything by the number in front of y

• y = (5 – 3x) ÷ 2
• y = 5/2 – 3x/2
• y = –3x/2 + 5/2

Or you can write it as: y = –1.5x + 2.5

Step 3: Now use a table of values to draw the graph

Worked Example:

Draw the graph of 3x + 2y = 5

After rearranging: y = –1.5x + 2.5

Calculations:

• When x = –2: y = –1.5(–2) + 2.5 = 3 + 2.5 = 5.5
• When x = 0: y = –1.5(0) + 2.5 = 2.5
• When x = 2: y = –1.5(2) + 2.5 = –3 + 2.5 = –0.5
• When x = 4: y = –1.5(4) + 2.5 = –6 + 2.5 = –3.5

Plot these points and draw the straight line.

Important Tips for Drawing Linear Graphs

1. Always use a ruler – linear graphs must be perfectly straight lines

2. Use a sharp pencil – this helps you plot points accurately

3. Label your axes – write "x" on the horizontal axis and "y" on the vertical axis

4. Choose a sensible scale – make sure your points will fit on the graph paper

5. Calculate at least 3 points – this helps you check your line is correct (if all three points don't line up, you've made a mistake)

6. Extend the line beyond your points – add small arrows at both ends to show the line continues forever

7. Check your work – pick a point on your line and check it satisfies the equation