Find the gradient of a line through two points practice questions | General Mathematics (Year 11, VCE Units 1 & 2, Linear functions and models, Straight-line graphs and slope)

Calculate a gradient

Gradient is vertical change divided by horizontal change. Use two distinct points on the same straight line

m=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1

From a graph Read the scale on each axis. Choose two clear grid points and count rise and run in coordinate units
From coordinates Subtract the vertical coordinates and the horizontal coordinates in the same point order. Divide the first difference by the second
Equivalent choices Reversing both differences leaves the gradient unchanged. Choosing another pair on the same line also gives the same gradient
Special cases Equal vertical coordinates give zero gradient when the points are distinct. Equal horizontal coordinates give undefined gradient

Find the gradient of the line through P and Q

\begin{aligned}P&=(-4,7)\\Q&=(2,-2)\end{aligned}

\(\displaystyle \Delta y=-2-7=-9\)Subtract the vertical coordinates from P to Q. The negative change describes a fall
\(\displaystyle \Delta x=2-(-4)=6\)Subtract the horizontal coordinates in the same order
\(\displaystyle m=\frac{-9}{6}\)Divide vertical change by horizontal change
\(\displaystyle m=-\frac{3}{2}\)Reduce the fraction by dividing numerator and denominator by their common factor

Reversing the ratio

✕\(\displaystyle m=\frac{\Delta x}{\Delta y}\)✓\(\displaystyle m=\frac{\Delta y}{\Delta x}\)

Gradient uses vertical change per horizontal unit, so vertical change belongs in the numerator

Reversing only one difference

✕\(\displaystyle m=\frac{y_2-y_1}{x_1-x_2}\)✓\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)

Both differences must describe the same movement between the points

Using one point alone

✕\(\displaystyle m=\frac{y_1}{x_1}\text{ for every line}\)✓\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)

A coordinate ratio works when the other point is the origin. In general, use the changes between two points

introFind a positive integer gradient from a unit grid

standardFind a signed integer or half-integer gradient from a grid or a coordinate pair

challengeFind exact fractional gradients from labelled graphs or point pairs, including horizontal lines

examChoose an exact gradient from a graph or two coordinates

Must I start at the left point?+

That makes the run positive and is often simpler. Starting at the right point also works if both differences use that same order

Why can I use another pair on the same straight line?+

The vertical and horizontal changes scale by the same factor. Their quotient stays the same

Why is a vertical gradient undefined?+

Two points on a vertical line have zero horizontal change. The gradient calculation would divide by zero

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 5: Linear functions and models