Gradient of a Line (Cambridge International IGCSE Mathematics)

Gradient of a line

What is the gradient of a line?

• The gradient is a measure of how steep a straight line is

• A gradient of 3 means:

  • For every 1 unit to the right, go up by 3

• A gradient of -4 means:

  • For every 1 unit to the right, go down by 4

• A gradient of 3 is steeper than 2

  • A gradient of -5 is steeper than -4

• A positive gradient means the line goes upwards (uphill)

  • Bottom left to top right

• A negative gradient means the line goes downwards (downhill)

• Top left to bottom right

How do I find the gradient of a line?

• Find two points on the line and draw a right-angled triangle

  • Then $\mathrm{gradient}=\frac{\mathrm{change} \mathrm{in}y}{\mathrm{change} \mathrm{in}x}$

  • Or, in short, $\frac{\mathrm{rise}}{\mathrm{run}}$

    • The rise is the vertical length of the triangle
    • The run is the horizontal length of the triangle

  • Put the correct sign on your answer

    • Positive for uphill lines
    • Negative for downhill lines

  • You can also find gradient of a line between two points, (x_ 1 comma y_ 1) and (x_ 2 comma y_ 2)

    • Use the formula $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

How do I draw a line with a given gradient?

• To draw the gradient $\frac{2}{3}$

  • The rise is 2

  • The run is 3

  • It is positive (uphill)

    • Move 3 units to the right and 2 units up

• To draw the gradient $-5$ make it a fraction, $-\frac{5}{1}$

  • The rise is 5

  • The run is 1

  • It is negative (downhill)

    • Move 1 unit to the right and 5 units down

Exam Hint A lot of students forget to make their gradients negative for downhill lines!

Worked Example (a) Find the gradient of the line shown in the diagram below.

Answer:

Find two points that the line passes through

(0 comma 2) and (1 comma 5)

Use the grid to draw a right-angled triangle Find the 'rise' (vertical length) and 'run' (horizontal length)

Work out the fraction $\frac{\mathrm{rise}}{\mathrm{run}}$

$\frac{3}{1}=3$

Look to see if the line is uphill or downhill

uphill, so the gradient is positive

Final answer: The gradient is 3

(b) On the grid below, draw the line with a gradient of −2 that passes through (0,1).

Answer:

Mark on the point (0, 1) -2 is the fraction $-\frac{2}{1}$ The rise is 2, the run is 1, the line goes downhill (so 1 across, 2 down)

(c) On the grid below, draw the line with a gradient of $\frac{2}{3}$ that passes through (0,-1).

Answer:

Mark on the point (0,-1) The rise is 2, the run is 3, the line goes uphill (so 3 across, 2 up)