Straight Line Graphs
Straight-line graphs are an essential part of Maths and appear in many problem-solving contexts. This guide will cover coordinates, equations of straight-line graphs, gradients, parallel and perpendicular lines, and finding the equation of a line through two points.
Coordinates
A coordinate system is used to locate points on a graph. The Cartesian coordinate system consists of two perpendicular axes:
- The $x$-axis (horizontal)
- The $y$-axis (vertical)
A point is written as ($x$, $y$), where:
- $x$ represents the horizontal position
- $y$ represents the vertical position
For example, the point ($3,5$) means:
- Move 3 units right on the $x$-axis
- Move 5 units up on the $y$-axis
The origin is the point ($0,0$), where the $x$-axis and $y$-axis intersect.
Straight Line Graphs
The equation of a straight line is generally written as:
$$y = mx + c$$
where:
- $m$ is the gradient (slope) of the line
- $c$ is the y-intercept (where the line crosses the $y$-axis)
Finding the Gradient
The gradient measures the steepness of a line and is calculated using two points:
$$m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
For example, given points ($1,2$) and ($4,8$):
$$m = \frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$
Interpreting the Gradient
- A positive gradient ($m>0$) means the line slopes upwards from left to right.
- A negative gradient ($m<0$) means the line slopes downwards from left to right.
- A zero gradient ($m=0$) means the line is horizontal.
- An undefined gradient means the line is vertical.
Finding the Y-Intercept
The y-intercept is the value of $y$ when $x$=0.
If the equation of the line is $y=3x+2y$, then the y-intercept is $c=2$.
Parallel and Perpendicular Lines
Parallel Lines
Parallel lines have the same gradient but different $y$-intercepts.
If a line has equation $y=2x+3$, any parallel line must have gradient $m=2$.
For example, $y=2x−5$ is parallel to $y=2x+3$.
Perpendicular Lines
Two lines are perpendicular if their gradients multiply to −1.
If a line has gradient $m$, then a perpendicular line has gradient:
$m_{\perp} = -\frac{1}{m}$
For example:
- If a line has equation $y = 3x + 2$, its gradient is 3.
- A perpendicular line must have a gradient of $-\frac{1}{3}$.
The Equation of a Line Through Two Points
If two points ($x1,y1$) and ($x2,y2$) are given, follow these steps to find the equation of the line:
Step 1: Find the Gradient
Use the formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Example: Find the equation of the line passing through ($2,3$) and ($6,11$).
$$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$$
Step 2: Use the Equation $y = mx + c$
Substitute one of the points into the equation to find $c$.
Using ($2,3$):
$$3 = 2(2) + c$$
$$3 = 4 + c$$
$$c = −1$$
Step 3: Write the Final Equation
$$y = 2x −1$$
This video shows you how to calculate the equation of a straight line given two points.
Exam Tips
✔ Always use the correct formula when calculating the gradient.
✔ Check if lines are parallel or perpendicular by comparing gradients.
✔ When finding the equation of a line, always substitute a point to find ccc.
✔ Plot points accurately when drawing graphs.
✔ Convert equations into the form $y = mx + c$ to identify gradient and y-intercept easily.
This guide provides the key concepts needed for straight-line graphs in GCSE Maths. Keep practising, and soon these techniques will feel easy.
