Proportion Graphs and Equations (Higher)

This lesson is Higher tier. It connects the algebra of proportion to its graphs. Being able to recognise direct and inverse proportion from the shape of a graph, read the constant of proportionality $k$k from it, and form the matching equation is a key skill in OCR GCSE Mathematics (J560). This lesson covers the straight-line graph of direct proportion, the curved graph of inverse proportion, forming equations from graph information, and interpreting points on proportion graphs.

This lesson develops AO1 technique in forming and using proportion equations, AO2 reasoning when you identify the type of proportion from a graph, and AO3 problem-solving when graph and equation information are combined.

Key Vocabulary

Term	Meaning
Direct proportion	$y = kx$y=kx; a straight line through the origin
Inverse proportion	$y = \dfrac{k}{x}$y=xk​; a curve falling away from both axes
Constant of proportionality ($k$k)	The fixed value linking the variables
Origin	The point $(0,0)$(0,0)
Gradient	The steepness of a straight line; for $y = kx$y=kx it equals $k$k
Asymptote	A line a curve approaches but never reaches (the axes, for $y = \frac{k}{x}$y=xk​)

Direct Proportion Graphs

If $y$y is directly proportional to $x$x, then $y = kx$y=kx. Its graph is a straight line through the origin, and the gradient of that line is the constant $k$k. The two features to recognise are therefore: it is straight, and it goes through $(0,0)$(0,0). The "through the origin" part is essential — a straight line that crosses the $y$y-axis somewhere other than zero is not direct proportion (it has the form $y = mx + c$y=mx+c with $c \ne 0$c=0), and the simple ratio $\frac{y}{x}$xy​ would not be constant for it. Doubling $x$x doubles $y$y, which is why the line is straight and passes through the origin.

Worked Example 1

A straight-line graph through the origin passes through the point $(4, 18)$(4,18). The relationship is $y = kx$y=kx. Work out $k$k and write the equation.

The gradient is $k = \dfrac{18}{4} = 4.5$k=418​=4.5.

So the equation is $y = 4.5x$y=4.5x.

Answer: $k = 4.5$k=4.5, equation $y = 4.5x$y=4.5x.

The gradient $4.5$4.5 tells you that every time $x$x increases by $1$1, $y$y increases by $4.5$4.5 — that constant, steady rate of increase is exactly what makes the line straight and gives direct proportion its distinctive graph.

Worked Example 1b

A direct-proportion graph passes through the origin and the point $(20, 8)$(20,8). Work out the equation of the line.

The gradient is $k = \dfrac{8}{20} = 0.4$k=208​=0.4.

So the equation is $y = 0.4x$y=0.4x.

Answer: $y = 0.4x$y=0.4x.

Common error: computing $\dfrac{20}{8} = 2.5$820​=2.5 instead. The gradient is "$y$y over $x$x" ($\dfrac{8}{20}$208​), so read the coordinates in the right order.

Worked Example 2

The graph of $y = kx$y=kx passes through $(6, 15)$(6,15). Use the equation to work out $y$y when $x = 10$x=10.

First find $k$k: $k = \dfrac{15}{6} = 2.5$k=615​=2.5, so $y = 2.5x$y=2.5x.

When $x = 10$x=10: $y = 2.5 \times 10 = 25$y=2.5×10=25.

Answer: $y = 25$y=25.

Common error: reading the gradient "upside down" as $\dfrac{x}{y}$yx​. The gradient (and $k$k) is always $\dfrac{y}{x}$xy​ for a point on $y = kx$y=kx.

Worked Example 2b

The cost $C$C of fabric is directly proportional to the length $L$L. A length of $6$6 m costs £15. Work out the cost of $10$10 m, and the length you could buy for £20.

Since cost is directly proportional to length, $C = kL$C=kL, where $k$k is the cost per metre: $k = \dfrac{15}{6} = 2.5$k=615​=2.5, i.e. £2.50 per metre. So $C = 2.5L$C=2.5L.

Cost of $10$10 m: $C = 2.5 \times 10 = 25$C=2.5×10=25, i.e. £25.

Length for £20: $20 = 2.5L$20=2.5L, so $L = \dfrac{20}{2.5} = 8$L=2.520​=8 m.

Answer: $10$10 m costs £25; £20 buys $8$8 m. This shows how a direct-proportion equation lets you work in either direction — cost from length, or length from cost.

Inverse Proportion Graphs

If $y$y is inversely proportional to $x$x, then $y = \dfrac{k}{x}$y=xk​. Unlike direct proportion, this graph is a curve, not a straight line. It falls steeply near the $y$y-axis and flattens towards the $x$x-axis, never quite touching either — lines that a curve approaches but never reaches are called asymptotes, and here both axes are asymptotes. The defining feature, and the one you use to identify or check inverse proportion, is that the product $x \times y = k$x×y=k is constant for every point on the curve. So if $(2, 12)$(2,12) is on the curve, then $(3, 8)$(3,8), $(4, 6)$(4,6) and $(6, 4)$(6,4) are too, because each has the same product.

Worked Example 3

A curve of the form $y = \dfrac{k}{x}$y=xk​ passes through $(5, 8)$(5,8). Work out $k$k and the equation.

For inverse proportion, $k = x \times y = 5 \times 8 = 40$k=x×y=5×8=40.

So the equation is $y = \dfrac{40}{x}$y=x40​.

Answer: $k = 40$k=40, equation $y = \dfrac{40}{x}$y=x40​.

The number $40$40 is the product that every point on this curve shares — for example $(5,8)$(5,8), $(8,5)$(8,5), $(10,4)$(10,4) and $(40,1)$(40,1) all give a product of $40$40.

Common error: dividing $\dfrac{8}{5}$58​ to find $k$k. For inverse proportion the constant is the product $xy$xy, not the quotient — that division would be the method for direct proportion.

Worked Example 3b

A curve $y = \dfrac{k}{x}$y=xk​ passes through the point $(8, 3)$(8,3). Work out the value of $x$x when $y = 6$y=6.

First find $k = xy = 8 \times 3 = 24$k=xy=8×3=24, so $y = \dfrac{24}{x}$y=x24​.

Now set $y = 6$y=6: $6 = \dfrac{24}{x}$6=x24​, so $x = \dfrac{24}{6} = 4$x=624​=4.

Answer: $x = 4$x=4. As a check, the product $4 \times 6 = 24$4×6=24 matches $k$k, confirming the point lies on the same curve.

Worked Example 4

The graph of $y = \dfrac{k}{x}$y=xk​ passes through $(2, 18)$(2,18). Use the equation to work out $y$y when $x = 12$x=12.

$k = 2 \times 18 = 36$k=2×18=36, so $y = \dfrac{36}{x}$y=x36​.

When $x = 12$x=12: $y = \dfrac{36}{12} = 3$y=1236​=3, found by substituting into the equation.

Answer: $y = 3$y=3. Notice that as $x$x increased from $2$2 to $12$12 (six times bigger), $y$y fell from $18$18 to $3$3 (six times smaller) — the hallmark of inverse proportion, where the two quantities change by reciprocal factors.

Worked Example 4b

The graph of $y = \dfrac{k}{x}$y=xk​ passes through $(9, 4)$(9,4). Work out $y$y when $x = 3$x=3.

Find the constant: $k = 9 \times 4 = 36$k=9×4=36, so $y = \dfrac{36}{x}$y=x36​.

When $x = 3$x=3: $y = \dfrac{36}{3} = 12$y=336​=12.

Answer: $y = 12$y=12. Dividing $x$x by $3$3 (from $9$9 to $3$3) multiplied $y$y by $3$3 (from $4$4 to $12$12), exactly as inverse proportion requires.

Recognising the Type from a Graph

A frequent exam task gives you a graph, a table, or a pair of points and asks you to decide whether the relationship is direct or inverse proportion. The reliable test is to check which constant rule holds: direct proportion has a constant ratio $\dfrac{y}{x}$xy​, while inverse proportion has a constant product $xy$xy. If neither is constant, the relationship is neither type of proportion. The table below summarises the features to look for.

Feature	Direct ($y = kx$y=kx)	Inverse ($y = \frac{k}{x}$y=xk​)
Shape	straight line	curve (reciprocal)
Through origin?	yes, passes through $(0,0)$(0,0)	no, never touches the axes
As $x$x increases	$y$y increases steadily	$y$y decreases towards $0$0
Constant rule	$\dfrac{y}{x} = k$xy​=k	$xy = k$xy=k

Worked Example 5

A graph passes through $(2, 12)$(2,12) and $(4, 6)$(4,6). State whether it shows direct or inverse proportion, and find $k$k.

Test the products: $2 \times 12 = 24$2×12=24 and $4 \times 6 = 24$4×6=24. The product $xy$xy is constant, so it is inverse proportion with $k = 24$k=24.

To be thorough, also check it is not direct proportion: $\dfrac{12}{2} = 6$212​=6 but $\dfrac{6}{4} = 1.5$46​=1.5, so the ratio is not constant, ruling out direct proportion.

The equation is $y = \dfrac{24}{x}$y=x24​.

Answer: inverse proportion, $k = 24$k=24, $y = \dfrac{24}{x}$y=x24​.

Common error: assuming any decreasing graph is inverse proportion. Only a curve with constant $xy$xy (and asymptotes at the axes) is true inverse proportion — a straight line sloping down is not.

Extended Worked Examples

Worked Example 6: Reading $k$k from a direct-proportion graph

A direct-proportion graph passes through $(8, 20)$(8,20). Work out the value of $y$y when $x = 14$x=14.