Straight-Line Graphs Revision Notes for Edexcel GCSE Maths

Straight-line graphs

Understanding straight-line graphs is a fundamental skill in coordinate geometry. These graphs represent linear relationships and have specific characteristics that make them easy to identify and work with.

Horizontal and vertical lines

Two of the most basic types of straight lines are horizontal and vertical lines, which have simple but important equations.

Vertical lines have the equation $x = a$ , where 'a' is any number. This creates a vertical line that passes through the point 'a' on the x-axis. For example, $x = 3$ creates a vertical line through the point $(3, 0)$ , and every point on this line has an x-coordinate of 3.

Horizontal lines have the equation $y = a$ , where 'a' is any number. This creates a horizontal line that passes through the point 'a' on the y-axis. For example, $y = 3$ creates a horizontal line through the point $(0, 3)$ , and every point on this line has a y-coordinate of 3.

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A common mistake students make is confusing $x = 3$ with $y = 3$ . Remember that $x = 3$ means all points have the same x-coordinate, whilst $y = 3$ means all points have the same y-coordinate.

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Worked Example: Identifying Points on Vertical and Horizontal Lines

For the vertical line $x = 5$ :

Points $(5, 1)$ , $(5, -2)$ , and $(5, 7)$ all lie on this line
Notice that the x-coordinate is always 5

For the horizontal line $y = -2$ :

Points $(1, -2)$ , $(4, -2)$ , and $(-3, -2)$ all lie on this line
Notice that the y-coordinate is always -2

The main diagonals

The main diagonal lines are special cases that pass through the origin and have slopes of exactly 1 or -1.

The line $y = x$ is known as the main diagonal that goes uphill from left to right. On this line, the x-coordinate and y-coordinate of each point are exactly the same.

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Worked Example: Points on the line $y = x$

Points like $(1, 1)$ , $(2, 2)$ , and $(-3, -3)$ all lie on this line because in each case, the y-coordinate equals the x-coordinate.

The line $y = -x$ is the main diagonal that goes downhill from left to right. On this line, the x-coordinate and y-coordinate of each point are negatives of each other.

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Worked Example: Points on the line $y = -x$

Points like $(1, -1)$ , $(2, -2)$ , and $(-3, 3)$ all lie on this line because in each case, the y-coordinate is the negative of the x-coordinate.

Other sloping lines through the origin

Lines that pass through the origin but have different slopes follow the general forms $y = ax$ and $y = -ax$ , where 'a' represents the gradient or steepness of the line.

The value of 'a' (called the gradient) determines how steep the line is. A larger value of 'a' creates a steeper line, whilst a smaller value creates a gentler slope. When the equation has a minus sign ( $y = -ax$ ), this tells you the line slopes downhill from left to right.

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These lines are particularly useful because they show direct proportional relationships between x and y values. This means that as x increases, y increases by a constant multiple.

Identifying straight lines from equations

Not all equations represent straight lines, so it's important to know how to identify them. All straight-line equations contain only three types of terms: something involving x, something involving y, and a number. These terms can be added or subtracted in any order.

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Worked Example: Identifying Straight-Line Equations

Examples of straight-line equations include:

$2y - 4x = 7$ (contains terms with y, x, and a number)
$6y - 8 = x$ (can be rearranged to $6y - x = 8$ )
$y = 3x + 2$ (contains terms with y, x, and a number)

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Examples of equations that are NOT straight lines include:

$y = x^2 + 3$ (contains $x^2$ )
$xy + 3 = 0$ (contains $xy$ )
$\frac{2}{y} - \frac{1}{x} = 7$ (contains fractions with variables in denominators)

The key is to look for terms like $x^2$ , $xy$ , or fractions where the variable appears in the denominator. These indicate that the equation represents a curved line rather than a straight line.

Working with coordinates

When working with straight-line graphs, understanding coordinates is essential. Each point on a line can be described using an $(x, y)$ coordinate pair, where x represents the horizontal position and y represents the vertical position.

For any straight line, you can find multiple coordinate pairs that satisfy the equation. This allows you to plot the line accurately on a coordinate grid and understand the relationship between the variables.

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Finding coordinate pairs is a key skill for graphing straight lines. You can substitute any value for x into the equation and calculate the corresponding y-value, or vice versa.

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Key Points to Remember:

Vertical lines have equation $x = a$ and pass through point $(a, 0)$
Horizontal lines have equation $y = a$ and pass through point $(0, a)$
Main diagonals are $y = x$ (uphill) and $y = -x$ (downhill)
Gradient determines steepness - larger values mean steeper lines
Straight-line equations only contain 'something x, something y and a number'

Straight-Line Graphs (Edexcel GCSE Maths): Revision Notes