Straight-line graphs 1 Revision Notes for Edexcel GCSE Maths

Straight-line graphs 1

When working with straight-line graphs, there are two essential things you need to understand. First, any equation written in the form $y = mx + c$ will always produce a straight line when graphed. Second, you can create these graphs systematically using a table of values.

Understanding the equation y = mx + c

The general form $y = mx + c$ contains two crucial components that determine how your straight line will look:

infoNote

In the equation $y = mx + c$ :

m represents the gradient (or slope) of the line
c represents the y-intercept - the point where the line crosses the y-axis

The gradient (m)

The gradient tells you how steep the line is and which direction it slopes. When the gradient is positive, the line slopes upwards from left to right. When it's negative, the line slopes downwards from left to right.

For every unit you move across (horizontally), the gradient tells you how many units you move up or down (vertically). For example, if the gradient is $-\frac{1}{2}$ , this means that for every unit you go across, you go half a unit down.

The y-intercept (c)

The y-intercept is where your line crosses the y-axis. This happens when $x = 0$ , so the coordinates of the y-intercept are always $(0, c)$ . This gives you an important starting point when drawing your graph.

Drawing graphs using a table of values

To draw a straight-line graph accurately, follow these steps:

Choose simple values of x - usually start with $-2, -1, 0, 1, 2$ to make calculations easier
Substitute each x-value into the equation to find the corresponding y-value
Create a table showing your x and y coordinates
Plot the points on your graph paper
Join the points with a straight line using a ruler

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

x	y
-1	-1
0	1
1	3
2	5

Calculations:

When $x = -1$ : $y = 2(-1) + 1 = -1$
When $x = 0$ : $y = 2(0) + 1 = 1$
When $x = 1$ : $y = 2(1) + 1 = 3$
When $x = 2$ : $y = 2(2) + 1 = 5$

Plot these coordinates and join them with a straight line.

Finding equations from graphs

If you're given a straight-line graph, you can work backwards to find its equation by identifying the gradient and y-intercept.

Finding the gradient

To find the gradient from a graph:

Draw a triangle on your line using two clear points
Calculate: gradient = rise ÷ run = change in y ÷ change in x
Remember: if the line slopes down from left to right, the gradient is negative

Finding the y-intercept

The y-intercept is simply where the line crosses the y-axis. Read this value directly from the graph as the coordinate $(0, c)$ .

Putting it together

Once you have both the gradient $(m)$ and y-intercept $(c)$ , substitute them into the general form $y = mx + c$ to get your equation.

For example, if the gradient is $5$ and the y-intercept is at $(0, 20)$ , then your equation is $y = 5x + 20$ .

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Worked Example: Rearranging $x + y = 4$

Sometimes equations aren't given in the standard $y = mx + c$ form. You can rearrange them:

$x + y = 4$ $y = -x + 4$

This gives us $m = -1$ and $c = 4$ . The gradient is $-1$ (line slopes downwards) and the y-intercept is at $(0, 4)$ .

When drawing this graph, it's still safer to create a table of values and plot at least three points before joining them with a straight line.

chatImportant

Exam Tips:

Always use a ruler to draw straight lines
Plot at least three points to ensure accuracy
When finding gradients from graphs, choose points that lie exactly on grid intersections
Check your y-intercept by seeing where the line crosses the y-axis
For negative gradients, remember the line goes "downhill" from left to right

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Key Points to Remember:

Any equation in the form $y = mx + c$ produces a straight line
The gradient $(m)$ tells you the slope direction and steepness
The y-intercept $(c)$ shows where the line crosses the y-axis
Use a table of values to plot points accurately before drawing your line
You can find an equation from a graph by identifying the gradient and y-intercept

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