Hard GCSE Rates of change on graphs Questions

Read what each axis shows

The gradient of the line is a rate: it is measured in the $y$-unit per the $x$-unit, so its units here are $\text{pounds per day}$.

Choose two points on the line

Any two points on a straight line give the same gradient, so use the two points that are stated exactly.

Work out the rise (the change in y)

The total hire cost changes by $60$ $\text{pounds}$ between the two points.

Work out the run (the change in x)

The time changes by $4$ $\text{days}$ over the same section.

Divide the rise by the run

The gradient of the line is $15$.

Attach the units to the gradient

A gradient without units is only half an answer: this rate is $15$ $\text{pounds per day}$.

Find where the line crosses the y-axis

The line cuts the $y$-axis at $50$, so there is a fixed amount of $50$ $\text{pounds}$ before the rate is applied.

Write the equation of the line

This is $y = mx + c$ with $m = 15$ and $c = 50$.

Check the equation with the other point

Substituting $x = 7$ gives $y = 155$, which matches the point $(7, 155)$, so the gradient and intercept are right.

Say what the gradient means here

For each extra $\text{day}$, the total hire cost goes up by $15$ $\text{pounds}$.

Avoid the classic error of turning the fraction upside down

Run divided by rise gives $\frac{1}{15}$ $\text{day per pound}$ — the rate the other way round, not what was asked for.

Check whether the graph shows direct proportion

The line is straight but it does NOT pass through the origin, so $y$ is NOT directly proportional to $x$ — the ratio $\frac{y}{x}$ changes from point to point.

Use the intercept to find the fixed charge

Substituting the point $(3, 95)$ into $y = mx + c$ gives $c = 50$. The intercept is the FIXED charge; the gradient is the RATE.

Keep the gradient and the intercept apart

The gradient $15$ is charged for every extra $\text{day}$; the intercept $50$ is paid once, whatever the time.

State the fixed charge

The fixed charge is $50$ $\text{pounds}$ — the value of $y$ when $x = 0$.

Read what each axis shows

Both gradients are rates in $\text{litres per minute}$, so they can be compared directly.

Work out the gradient of line A

Line A passes through the origin, so its rate is $2.25$ $\text{litres per minute}$.

Work out the gradient of line B

Line B passes through the origin, so its rate is $3.2$ $\text{litres per minute}$.

Compare the two gradients

Line B is the steeper line, so line B has the greater rate of change.

Work out the difference between the rates

The two rates differ by $0.95$ $\text{litres per minute}$.

Rule out the "same rate" option

The gradients are different, so the lines are not parallel and the rates are not equal.

Rule out the option that names the wrong line

Line A is the shallower line, so it cannot have the greater rate of change.

Rule out the options with the wrong difference

Only $0.95$ is the true difference between the two gradients.

Check line A with its point

The gradient rebuilds the point $(8, 18)$, so $m_A$ is right.

Check line B with its point

The gradient rebuilds the point $(5, 16)$, so $m_B$ is right.

Avoid comparing the two named points directly

The two points are at different $x$-values, so their $y$-values cannot be compared. Only the gradients can.

Interpret the difference in context

Every $\text{minute}$, line B gains $0.95$ $\text{litres}$ more than line A.

Note both lines show direct proportion

Both lines are straight and pass through the origin, so both graphs show direct proportion.

Reflect on the method

On the same axes, the steeper line always has the greater rate of change.

Choose the correct statement

Line B has the greater rate of change, by $0.95$ $\text{litres per minute}$.

Recall what a direct-proportion graph looks like

Direct proportion gives a STRAIGHT LINE THROUGH THE ORIGIN. Doubling $x$ doubles $y$, and $\frac{y}{x}$ is the same at every point.

Recall what an inverse-proportion graph looks like

Inverse proportion gives a CURVE (a hyperbola) that falls steeply and then flattens, approaching both axes without touching them. Doubling $x$ halves $y$, and $xy$ is the same at every point.

Apply the test to each option

For each option, check the ratio $\frac{y}{x}$ (direct) and the product $xy$ (inverse). Only one option passes the test that was asked for.

Rule out straight lines that miss the origin

A straight line that crosses the $y$-axis away from the origin is NOT proportional: at $x = 0$ the value of $y$ is not $0$, so $\frac{y}{x}$ is not constant.

Rule out horizontal lines

A horizontal line keeps $y$ fixed while $x$ changes, so neither $\frac{y}{x}$ nor $xy$ is constant. It shows no proportional relationship at all.

Rule out curves that are not hyperbolas

A U-shaped curve through the origin has $y$ proportional to $x^2$, not to $x$, and $xy$ is not constant either. Points such as $(1, 2)$, $(2, 8)$ and $(3, 18)$ fit $y = 2x^2$.

Check a listed set of points for direct proportion

The points $(1, 3)$, $(2, 6)$, $(3, 9)$ all give $\frac{y}{x} = 3$, so they lie on $y = 3x$ — a straight line through the origin.

Check a listed set of points for inverse proportion

The points $(1, 12)$, $(2, 6)$, $(3, 4)$ all give $xy = 12$, so they lie on the curve $y = \frac{12}{x}$ — inverse proportion.

Check the points that only look proportional

The points $(1, 5)$, $(2, 7)$, $(3, 9)$ lie on the straight line $y = 2x + 3$. The ratio $\frac{y}{x}$ changes, so this is not direct proportion.

Say what the gradient means for a proportional graph

On a direct-proportion graph the gradient IS the constant of proportionality — and it is the rate of change of $y$ with respect to $x$.

Note the rate of change on an inverse graph

A hyperbola has a different gradient at every point, so its rate of change is not constant — unlike a straight line.

Watch the axes

The single quickest test for direct proportion is whether the graph goes through the origin. If it does not, it cannot be proportional.

Compare the two shapes side by side

One is a straight line through the origin; the other is a curve that never touches the axes. They are never the same graph.

Reflect on the question asked

The question asked for INVERSELY proportion, so an direct-proportion graph, however tempting, is the wrong answer.

Choose the correct graph

The hyperbola is the only graph on which $xy$ is constant.

Read what each axis shows

The gradient of the line is a rate: it is measured in the $y$-unit per the $x$-unit, so its units here are $\text{litres per minute}$.

Choose two points on the line

Any two points on a straight line give the same gradient, so use the two points that are stated exactly.

Work out the rise (the change in y)

The volume of oil in the barrel changes by $-90$ $\text{litres}$ between the two points.

Work out the run (the change in x)

The time changes by $30$ $\text{minutes}$ over the same section.

Divide the rise by the run

The gradient of the line is $-3$.

Attach the units to the gradient

A gradient without units is only half an answer: this rate is $-3$ $\text{litres per minute}$.

Find where the line crosses the y-axis

The line cuts the $y$-axis at $90$, so there is a fixed amount of $90$ $\text{litres}$ before the rate is applied.

Write the equation of the line

This is $y = mx + c$ with $m = -3$ and $c = 90$.

Say what the gradient means here

For each extra $\text{minute}$, the volume of oil in the barrel goes down by $3$ $\text{litres}$.

Avoid the classic error of turning the fraction upside down

Run divided by rise gives $-\frac{1}{3}$ $\text{minute per litre}$ — the rate the other way round, not what was asked for.

Test whether the line passes through the origin

When $x = 0$, $y = 90$, not $0$. The line misses the origin, so $y$ is NOT directly proportional to $x$ — even though it is perfectly straight.

Test whether the ratio y over x is constant

Direct proportion needs $\frac{y}{x}$ to be the same at every point. Here it changes from point to point, confirming that $y$ is not directly proportional to $x$.

Test whether the product xy is constant

Inverse proportion would need $xy$ to be constant, and it would need a curved graph. A straight line can never show inverse proportion.

Rule out the "curve" option

The graph is described as a straight line, so any statement that calls it a curve is false.

Choose the correct statement

The line is straight but does not pass through the origin, so $y$ is not directly proportional to $x$.

Read what each axis shows

The gradient of the line is a rate: it is measured in the $y$-unit per the $x$-unit, so its units here are $\text{pounds per cubic metre}$.

Choose two points on the line

Any two points on a straight line give the same gradient, so use the two points that are stated exactly.

Work out the rise (the change in y)

The total cost changes by $24$ $\text{pounds}$ between the two points.

Work out the run (the change in x)

The water used changes by $6$ $\text{cubic metres}$ over the same section.

Divide the rise by the run

The gradient of the line is $4$.

Attach the units to the gradient

A gradient without units is only half an answer: this rate is $4$ $\text{pounds per cubic metre}$.

Find where the line crosses the y-axis

The line cuts the $y$-axis at $13$, so there is a fixed amount of $13$ $\text{pounds}$ before the rate is applied.

Write the equation of the line

This is $y = mx + c$ with $m = 4$ and $c = 13$.

Say what the gradient means here

For each extra $\text{cubic metre}$, the total cost goes up by $4$ $\text{pounds}$.

Avoid the classic error of turning the fraction upside down

Run divided by rise gives $0.25$ $\text{cubic metre per pound}$ — the rate the other way round, not what was asked for.

Find the y-intercept

Substituting $(3, 25)$ into $y = mx + c$ gives $c = 13$: the value of $y$ when $x = 0$.

Say what the intercept means here

It is the FIXED charge of $13$ $\text{pounds}$, paid before any water used at all.

Keep the intercept and the gradient apart

The gradient $4$ is the RATE, charged per $\text{cubic metre}$; the intercept $13$ is the one-off fixed amount. Swapping them is the usual mistake.

Rule out "no fixed charge"

The line does not pass through the origin, so there certainly is a fixed charge.

Choose the correct interpretation

The intercept is the fixed charge of $13$ $\text{pounds}$.