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The Straight Line

A straight line is the shortest distance between two points.

The gradient of a line is a measure of its slope relative to the x‑axis.

straight line

A positive gradient slopes upwards from left to right.

slope up positive gradient

A negative gradient slopes downwards from left to right.

negative gradient slope down

Gradient

The gradient is found by dividing the vertical distance by the horizontal distance.

\[ \text{Gradient } = \frac{\text{Vertical distance}}{\text{Horizontal distance}} \] \[ \text{This is often shortened to} \] \[ \qquad m = \frac{V}{H} \] \[ \text{where } m = \text{gradient},\; V = \text{Vertical distance},\; H = \text{Horizontal distance} \]
\[ \text{Since the vertical distance corresponds to the } y \text{ direction,} \] \[ \text{and the horizontal distance corresponds to the } x \text{ direction,} \] \[ \qquad m = \frac{y}{x} \]
Example
slope up

The horizontal distance is 2 units. The vertical distance is 4 units.

\[ m = \frac{V}{H} = \frac{4}{2} = 2 \]

The gradient is 2.

Gradient Triangle

The gradient triangle can be used to find a vertical, horizontal, or gradient.

v=mH triangle
Examples

Find gradient given vertical and horizontal

slope up
\[ m = \frac{V}{H} \] \[ = \frac{3}{6} \] \[ = \frac{1}{2} \]
slope down
\[ m = \frac{V}{H} \] \[ = \frac{4}{-2} \] \[ = -2 \]

vertical
\[ m = \frac{V}{H} \] \[ = \frac{5}{0} \] \[ = \text{No gradient} \]
horizontal
\[ m = \frac{V}{H} \] \[ = \frac{0}{8} \] \[ = 0 \]
Example

Find horizontal given vertical and gradient

slope up given gradient
\[ m = \frac{V}{H} \] \[ H = \frac{V}{m} \] \[ x = \frac{20}{4} \] \[ x = 5 \text{ units} \]
Example

Find vertical given horizontal and gradient

slope up find side
\[ m = \frac{V}{H} \] \[ mH = V \] \[ 2 \times 7 = x \] \[ 14 = x \] \[ x = 14 \text{ units} \]
Gradient formula

Drawing a Line from its Equation

To draw a line from its equation:

  • Draw a table with x and y.
  • Pick values for x.
  • Substitute each x into the equation to find y.
  • Plot the points (x, y).
Example
drawing a line animation

This gives the points:

(0,0), (1,2), (2,4), (3,6), (4,8)

Plotting these points gives:

y=2x

Extrapolating backwards:

y=2x values y=x
Example

The line y = x has values:

y=x

And graph:

y=x

Harder lines

  • Draw a table with x and y.
  • Pick values for x.
  • Multiply x by the coefficient, then add/subtract the constant.
  • Fill in y.
  • Plot the points.
Example
y=2x+3

The line y = 2x + 3

y=2x+3

The Equation of a Straight Line

y=mx+c
\[ y = mx + c \]

Where \( m \) = gradient and \( c \) = y‑intercept (the value of \( y \) when \( x = 0 \)).

Example
y=2x+1
Plotting the graph Finding the equation Straight Line – Part 2