Straight-line graphs 1 Revision Notes for AQA GCSE Maths

Straight-line graphs 1

Understanding linear equations

All equations written in the form $y = mx + c$ produce straight-line graphs when plotted. This is the most important form to remember for linear equations.

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The Standard Form: $y = mx + c$

In this equation:

m represents the gradient (how steep the line is)
c represents the y-intercept (where the line crosses the y-axis)

This form allows you to immediately identify the key characteristics of any straight line.

The gradient (m)

The gradient tells you how much the line slopes up or down. It shows the rate of change between x and y values.

A positive gradient means the line slopes upwards from left to right
A negative gradient means the line slopes downwards from left to right
The larger the number, the steeper the slope

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Understanding Gradient: $y = -\frac{1}{2}x + 5$

In this equation:

The gradient is $-\frac{1}{2}$
This means for every 1 unit you move right, you move $\frac{1}{2}$ unit down
The negative sign indicates a downward slope from left to right

The y-intercept (c)

The y-intercept is the point where your straight line crosses the y-axis. This always happens when $x = 0$ .

In $y = -\frac{1}{2}x + 5$ :

The y-intercept is 5
The line crosses the y-axis at the point $(0, 5)$

Drawing graphs using table of values

When you need to plot a straight-line graph, creating a table of values is a reliable method that works for any linear equation.

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Why Use a Table of Values?

A table of values provides multiple coordinate points that you can plot and connect. This method is particularly useful when the gradient or y-intercept isn't immediately obvious from the equation's form.

Steps to follow:

Choose simple x values - usually start with $-2, -1, 0, 1, 2$
Substitute each x value into your equation to find the corresponding y value
Create coordinate pairs from your x and y values
Plot these points on your graph
Join the points with a straight line using a ruler

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Worked Example: Plotting $y = 2x + 1$

Step 1: Create your table of values

When $x = 0$ : $y = 2(0) + 1 = 1$ , giving point $(0, 1)$
When $x = 1$ : $y = 2(1) + 1 = 3$ , giving point $(1, 3)$
When $x = 2$ : $y = 2(2) + 1 = 5$ , giving point $(2, 5)$

Step 2: Plot these coordinate pairs on your graph

Step 3: Connect the points with a straight line using a ruler

Finding equations from graphs

Sometimes you'll need to work backwards from a graph to find its equation. This involves identifying the gradient and y-intercept from the visual information.

Finding the gradient

To calculate the gradient from a graph, use the rise over run method:

Draw a triangle between two clear points on the line
Count the vertical change (rise)
Count the horizontal change (run)
Calculate: Gradient = $\frac{\text{rise}}{\text{run}}$

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Direction Matters for Gradient

Remember to consider the direction:

If the line goes up from left to right, the gradient is positive
If the line goes down from left to right, the gradient is negative

Finding the y-intercept

Look at where your straight line crosses the y-axis. The y-coordinate of this point is your y-intercept (c value).

Writing the equation

Once you have both values:

Put the gradient in place of m
Put the y-intercept in place of c
Write your equation in the form $y = mx + c$

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Worked Example: Finding an Equation from a Graph

Looking at a line on a graph:

Step 1: Calculate the gradient

The gradient triangle shows a rise of 20 and a run of 4
Gradient = $\frac{20}{4} = 5$

Step 2: Identify the y-intercept

The y-intercept is at $(0, 20)$

Step 3: Write the equation

Therefore the equation is $y = 5x + 20$

Rearranging equations

Some equations might not be given in the standard $y = mx + c$ form. You can rearrange them to make graphing easier.

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Making Equations Graph-Ready

Not all linear equations are presented in the convenient $y = mx + c$ form. Rearranging helps you quickly identify the gradient and y-intercept for easy graphing.

For example: $x + y = 4$

Subtract x from both sides: $y = -x + 4$
Now you can see the gradient is $-1$ and the y-intercept is $4$

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Key Points to Remember:

All straight-line equations can be written in the form $y = mx + c$
The gradient (m) shows how steep the line is and which direction it slopes
The y-intercept (c) is where the line crosses the y-axis
Table of values is a reliable method for plotting any linear graph
Always use a ruler when drawing straight lines on graphs
Check your work by ensuring all plotted points lie on your straight line

Straight-line graphs 1 (AQA GCSE Maths): Revision Notes