Gradient and y-intercept — what do they mean — Mathematics

Before this

Gradient — steepness and direction · coming soon
Drawing straight line graphs from an equation
Simultaneous equations word problems

This leads to

Finding the equation of a straight line
Displacement-time graphs (Physics)
Differentiation from first principles · coming soon

Two graphs of the same taxi

Two taxi drivers in the same city each show you a graph of their fares. Driver A's line climbs steeply. Driver B's line is almost flat. Which driver charges more for each kilometre?

Two graphs of fares. Look at the numbers on the cost axes before you decide which line is "steeper".

Most people choose Driver A, because a steeper line looks more expensive. Now read the axes. Both lines start at $2.50. Both reach $20.50 at 10 km. So the two drivers charge exactly the same fare: $2.50 to get in, plus $1.80 for every kilometre. Driver B simply drew the cost axis to $100 instead of $25, which squashes the line.

This is the disruption. The steepness your eye sees depends on how the graph was drawn. The gradient does not. The gradient is a number, and in a real situation it has units: here, dollars per kilometre.

The last lecture showed how to draw $y = m x + c$ , and promised to explore what the gradient means. This lecture keeps that promise. It also asks what the other number, $c$ , means. By the end, the two numbers in $y = m x + c$ will read like a sentence about the world.

c is the value of y when x is zero

Start with the easier number. Put $x = 0$ into $y = m x + c$ :

y = m (0) + c = c

So $c$ is the value of $y$ when $x$ is zero. On a graph, the points with $x = 0$ form the $y$ -axis. So the line crosses the $y$ -axis at $(0, c)$ . This is why $c$ is called the $y$ -intercept.

In a real situation, $x = 0$ usually means "before anything has happened". So $c$ is the starting value. Suppose a phone contract costs $40 to set up, plus $30 for each month. Let $x$ be the number of months and $y$ the total cost in dollars:

y = 30 x + 40

At $x = 0$ , you have used the phone for no months. But you have already paid $40. So the line starts at $(0, 40)$ , above the origin.

Notice the word "usually". The value $x = 0$ means "the start" only when $x$ counts from the start. If $x$ were the calendar year, then $x = 0$ would be the year 0, which means nothing for a phone contract. So before you interpret $c$ , always ask what $x = 0$ means in the story.

m is a rate of change

Now the richer number. In the last lecture, the gradient was the amount added to $y$ for each step of 1 in $x$ . For the phone contract, each extra month adds $30. So the gradient, 30, is the monthly charge.

The phone contract. A step of 1 month adds $30. A step of 3 months adds $90. Both steps give

90 \div 3 = 30 \div 1 = 30

dollars per month.

But real data rarely comes in steps of 1. Take the points for month 2 and month 5. Between them, $x$ changes by $5 - 2 = 3$ , and $y$ changes by $190 - 100 = 90$ . Divide the change in $y$ by the change in $x$ : $90 \div 3 = 30$ . The same 30 appears. This gives the general definition.

The gradient of a straight line

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on a line,

$m = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

The Greek letter $Δ$ (delta) means "change in". The gradient is the change in $y$ for each 1 unit of change in $x$ .

Why do any two points give the same answer? The last lecture showed that on a straight line, every step of 1 in $x$ adds the same amount to $y$ . Here that amount is 30. A change of 3 in $x$ is three such steps, so it adds $3 \times 30 = 90$ . In general, the change in $y$ is always 30 times the change in $x$ . So their ratio is always 30, whichever two points you choose. In the figure, the two dashed triangles have different sizes but exactly the same shape.

Now read the definition slowly: change in $y$ per change in $x$ . A quantity "per" another quantity is a rate. And each change carries its own units:

m = \frac{90 dollars}{3 months} = 30 dollars per month

So in a real situation, the gradient has units: dollars per month, metres per second, degrees per kilometre. The units are always "the units of $y$ per the units of $x$ ". When both axes use the same unit, the units cancel. A ramp that rises 1 m for every 12 m across has gradient $\frac{1}{12}$ , a pure number.

That is it. That is the whole idea. The gradient is not a steepness to look at. It is a rate to read: how fast $y$ changes as $x$ changes. Go back to the taxis. Each graph has a rise of $20.5 - 2.5 = 18$ dollars over a run of 10 km. Both gradients are $18 \div 10 = 1.80$ dollars per kilometre. The drawing changed the look of the line, not the rate.

Stop and think

Find the gradient through $(1, 70)$ and $(4, 160)$ in two ways: as $\frac{160 - 70}{4 - 1}$ , and as $\frac{70 - 160}{1 - 4}$ . Does the order matter? What would happen if you used $160 - 70$ on top and $1 - 4$ underneath?

Show the answer

The first way gives $\frac{90}{3} = 30$ . The second gives $\frac{- 90}{- 3} = 30$ . The order does not matter, because swapping the points changes the sign of the top and the bottom, and the two signs cancel. The danger is mixing the orders. With $160 - 70$ on top and $1 - 4$ underneath, you get $\frac{90}{- 3} = - 30$ : the right size with the wrong sign. The rule is simple: whichever point comes first on top must also come first underneath.

Here is that exact mistake, hidden in otherwise careful working.

Spot the error — the gradient through two points

Amara finds the gradient of the line through A(−1, 4) and B(3, −4). One line of her working is wrong. Tap it.

Amara's working for the gradient through A $(- 1, 4)$ and B $(3, - 4)$ , with the error marked:

Change in $y$ , from A to B: $- 4 - 4 = - 8$ . Correct.
Change in $x$ : $- 1 - 3 = - 4$ . Wrong: this goes from B to A. From A to B it is $3 - (- 1) = 4$ .
Gradient $= - 8 \div (- 4) = 2$ . The division is right, but it uses the wrong $Δ x$ .
Correct answer: $m = - 8 \div 4 = - 2$ . As $x$ goes up, $y$ goes down, so a negative gradient makes sense.

The quickest check is a picture in your head. If $y$ goes down while $x$ goes up, the gradient must be negative. A sign that disagrees with the picture means the subtraction orders do not match.

Up, down or flat: what the sign tells you

The sign of $m$ tells you the direction of change.

Positive gradient, $m > 0$ . As $x$ increases, $y$ increases. The line rises from left to right. In a story, something is growing: savings, a total bill, water in a tank that is filling.
Negative gradient, $m < 0$ . As $x$ increases, $y$ decreases. The line falls from left to right. Something is being used up: a battery draining, a candle burning down, the distance left to travel.
Zero gradient, $m = 0$ . The equation becomes $y = c$ . The line is flat, because $y$ does not change at all. A library card that costs $15 a year, however many books you borrow, has a zero gradient against books borrowed.

What about a vertical line? For any two points on it, $x_{2} - x_{1} = 0$ . So the gradient formula would divide by zero, and that has no answer. A vertical line has no gradient: we say its gradient is undefined. This is the gradient's view of a fact from the last lecture. A vertical line cannot be written as $y = m x + c$ , because there is no number $m$ to put in.

The size of $m$ , ignoring its sign, tells you how fast the change happens. A line with gradient $- 3$ changes faster than a line with gradient 2, just in the downward direction. On the same axes, a phone plan at $50 per month is drawn steeper than one at $20 per month. On the same axes, steeper really does mean faster.

Now take the line apart yourself. Change $m$ and $c$ one at a time, and watch what each one does.

Graph builder — what m and c each do

Change m and c with the sliders or the buttons. First, change only c: what happens to the line? Then change only m. Then try three tasks. Task 1: make the line pass through (0, −1) and (3, 5). Task 2: make a line that falls 1.5 for every 1 across and crosses the y-axis at 4. Task 3: set m = 0.

Gradient m = 1

y-intercept c = 2

Three lines with different gradients. The sign of $m$ decides the direction, and $c$ decides where each line crosses the $y$ -axis.

y = 2 x + 1

rises 2 for every 1 across and crosses the

y

-axis at

(0, 1)

y = - x + 3

falls 1 for every 1 across and crosses at

(0, 3)

y = 2

is flat, with gradient 0. Changing

c

alone slides a line up or down. Changing

m

alone turns it about the point

(0, c)

What did you see? Changing $c$ slides the whole line up or down without turning it. Changing $m$ turns the line, and the point $(0, c)$ stays fixed while it turns. The two numbers do two separate jobs: $c$ fixes one point, and $m$ fixes the direction. This is why two numbers are enough to describe any straight line that is not vertical.

Worked examples

Worked Example 1 — The taxi fare, read as a sentence

A taxi charges a booking fee of $2.50, plus $1.80 per kilometre. Write an equation for the cost $C$ dollars of a journey of $d$ km. State and interpret the gradient and the intercept.

Equation. Each kilometre adds $1.80, so $d$ kilometres add $1.80 d$ . The fee is added once. So $C = 1.80 d + 2.50$ . This is $y = m x + c$ with $C$ in place of $y$ and $d$ in place of $x$ .

Gradient. $m = 1.80$ dollars per kilometre: each extra kilometre adds $1.80 to the fare.

Intercept. $c = 2.50$ dollars: the cost at $d = 0$ , before the taxi moves. It is the booking fee.

Where the model makes sense. The line goes on forever in both directions, but a distance cannot be negative. So only the part with $d \geq 0$ describes real journeys. Always ask which part of the line is meaningful in the story.

Check. At $d = 10$ : $1.80 \times 10 + 2.50 = 20.50$ , the point $(10, 20.5)$ on both drivers' graphs ✓.

Worked Example 2 — Reading m and c from a graph

A line crosses the $y$ -axis at $(0, - 1)$ and passes through $(3, 5)$ . Find its gradient and intercept, and write its equation.

Intercept. The line crosses the $y$ -axis at $(0, - 1)$ , so $c = - 1$ . A negative intercept is completely normal. It means the line crosses below the origin, or in a story, that the starting value is below zero.

Gradient. Use the two points, in the same order on top and underneath:

m = \frac{5 - ( - 1 )}{3 - 0} = \frac{6}{3} = 2

Equation. $y = 2 x - 1$ .

Check. A third point on the drawn line, such as $(1, 1)$ : $2 (1) - 1 = 1$ ✓. This method needs the intercept to be visible on the graph. The next lecture finds the equation when it is not.

Worked Example 3 — Two tariffs that swap over

Two phone tariffs cost $y$ dollars after $x$ months. Tariff A: $y = 15 x + 10$ . Tariff B: $y = 12 x + 25$ . Which is cheaper for 3 months? For 10 months? When do they cost the same?

3 months. A: $15 (3) + 10 = 55$ . B: $12 (3) + 25 = 61$ . A is cheaper.

10 months. A: $15 (10) + 10 = 160$ . B: $12 (10) + 25 = 145$ . B is cheaper.

Read the two numbers. B starts $15 higher (intercepts 25 and 10). But B charges $3 less each month (gradients 12 and 15). So B closes the gap by $3 every month. A gap of $15, closed at $3 per month, takes $15 \div 3 = 5$ months.

Check by algebra. The costs are equal when $15 x + 10 = 12 x + 25$ . Then $3 x = 15$ , so $x = 5$ . At $x = 5$ , A costs $15 (5) + 10 = 85$ and B costs $12 (5) + 25 = 85$ ✓. This is a pair of simultaneous equations, and the crossing point is its solution.

Tariff A starts lower but rises faster. The lines cross at

(5, 85)

: before 5 months A is cheaper, and after 5 months B is cheaper.

Worked Example 4 — A falling line: a draining battery

A phone's battery shows 92% at 08:00 and 60% at 12:00. Assume it drains at a steady rate. Write an equation for the charge $y$ (in %) after $t$ hours from 08:00. When will the battery be empty?

Choose what $t = 0$ means. Counting from 08:00 makes the intercept the 08:00 reading: $c = 92$ .

Gradient. From $(0, 92)$ to $(4, 60)$ : $m = \frac{60 - 92}{4 - 0} = \frac{- 32}{4} = - 8$ . The units are percentage points per hour. The minus sign is information: the charge is being used up.

Equation. $y = - 8 t + 92$ .

Empty. Put $y = 0$ : $0 = - 8 t + 92$ , so $8 t = 92$ and $t = 11.5$ . That is 11.5 hours after 08:00, which is 19:30.

Check. At $t = 4$ : $- 32 + 92 = 60$ ✓. The model only makes sense for $0 \leq t \leq 11.5$ . A real battery does not drain at a perfectly steady rate, so the line is a model, and 19:30 is an estimate.

Three classic mistakes

Trap 1 — Judging the gradient by eye

A line can look steep or flat depending on the scales of the axes, as the two taxi graphs showed. Never compare gradients by looking at lines on different graphs. Read two points, calculate $Δ y \div Δ x$ , and write the units.

Trap 2 — Mixing the order, or turning it upside down

The gradient is $Δ y$ on top and $Δ x$ underneath, never the other way round. And whichever point comes first on top must come first underneath. If the line falls from left to right, your answer must be negative. Use that as a check every time.

Trap 3 — Reading c where the graph begins

The intercept is the value of $y$ where $x = 0$ . Some graphs start their $x$ -axis at another value, such as 2000 for a year, or 5 for a temperature. Then the left edge of the graph is not the $y$ -axis, and the value there is not $c$ . Check that the $x$ -axis really begins at 0 before you read an intercept.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck. All situations are invented.

A candle's height $h$ cm after $t$ hours of burning is $h = 24 - 1.5 t$ . State the gradient and the intercept, with units, and say what each means. How tall is the candle after 10 hours? When does it burn out? For which values of $t$ does the model make sense?

Hint 1
Rewrite it as $h = - 1.5 t + 24$ , so the number multiplying $t$ is easy to see.

Hint 2
"Burns out" means $h = 0$ . Solve $0 = 24 - 1.5 t$ .

Full answer
Gradient $- 1.5$ cm per hour: the candle gets 1.5 cm shorter each hour. Intercept 24 cm: its height before it is lit. After 10 hours: $24 - 1.5 \times 10 = 24 - 15 = 9$ cm. It burns out when $1.5 t = 24$ , so $t = 16$ hours. Check: $24 - 1.5 \times 16 = 0$ ✓. The model makes sense for $0 \leq t \leq 16$ . After 16 hours there is no candle, so a negative height means nothing.
Find the gradient of the line through $(- 2, 7)$ and $(4, - 5)$ . Do it twice, once with each point first, and show that you get the same answer. Then check your answer using the point $(1, 1)$ , which is also on the line.

Hint 1
Write $\frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ and keep the same point first on top and underneath. Brackets help with the negatives.

Hint 2
First way: $\frac{- 5 - 7}{4 - ( - 2 )}$ . Second way: $\frac{7 - ( - 5 )}{- 2 - 4}$ .

Full answer
First way: $\frac{- 12}{6} = - 2$ . Second way: $\frac{12}{- 6} = - 2$ . The same, because both signs flipped. Check with $(- 2, 7)$ and $(1, 1)$ : $\frac{1 - 7}{1 - ( - 2 )} = \frac{- 6}{3} = - 2$ ✓. Any two points on a straight line give the same gradient. The line falls as $x$ increases, so a negative gradient fits the picture.
Two gyms. Gym P charges a $60 joining fee and $25 per month. Gym Q has no joining fee and charges $35 per month. Write an equation for each. Which is cheaper for 3 months, and for 12 months? After how many months do they cost the same? Explain the answer using only the gradients and the intercepts.

Hint 1
P: $y = 25 x + 60$ . What is Q's intercept?

Hint 2
Q starts $60 cheaper. Each month, how much of that advantage does Q lose?

Full answer
P: $y = 25 x + 60$ . Q: $y = 35 x$ , with intercept 0. For 3 months: P costs $135$ dollars and Q costs $105$ , so Q is cheaper. For 12 months: P costs $25 \times 12 + 60 = 360$ and Q costs $35 \times 12 = 420$ , so P is cheaper. Equal when $25 x + 60 = 35 x$ , so $10 x = 60$ and $x = 6$ months, when both cost $210. Using the two numbers: Q starts $60 cheaper, but it charges $10 more per month, so its advantage is gone after $60 \div 10 = 6$ months ✓.
Two runners plot distance against time. Ravi's graph has time in minutes and distance in metres, and its gradient is 250. Mei's graph has time in seconds and distance in metres, and its gradient is 4. Who runs faster? Explain why comparing the numbers 250 and 4 directly would be wrong.

Hint 1
Write each gradient with its units: "metres per minute" and "metres per second". Units are part of the answer.

Hint 2
Convert both to the same unit, such as km per hour. There are 60 minutes in an hour, 3600 seconds in an hour, and 1000 m in a km.

Full answer
Ravi: 250 m per minute is $250 \times 60 = 15000$ m per hour, which is 15 km/h. Mei: 4 m per second is $4 \times 3600 = 14400$ m per hour, which is 14.4 km/h. So Ravi is slightly faster. Comparing 250 with 4 directly would be wrong because they are rates in different units. A gradient is a number with units, and two rates can be compared only in the same units. This is Trap 1 in another form: the number, like the look of a line, depends on how the graph was set up. Check: Mei's speed in metres per minute is $4 \times 60 = 240$ , a little less than Ravi's 250 ✓.

The bigger idea

A straight line has one gradient everywhere. That is what makes it straight: the rate of change never changes. But most things in the world do not change at a steady rate. A ball thrown upwards slows down, stops and falls. A population grows faster as it gets bigger.

For a curve, the gradient is different at every point. So a new question appears: how fast is $y$ changing at this exact point? Answering that question is the job of differentiation, the first half of calculus. The idea behind it is the one in this lecture. Take two points, find $Δ y \div Δ x$ , and then move the two points closer and closer together.

The same idea already runs through other subjects. The gradient of a distance–time graph is a speed. The gradient of a velocity–time graph is an acceleration. In Economics, the gradient of a cost line is the cost of one more unit. Wherever one quantity depends on another, the gradient tells you how strongly one responds to the other. Learning to read it as a rate, with units, is learning to read graphs as statements about the world.

Key concepts in this lesson

In $y = m x + c$ , the intercept $c$ is the value of $y$ when $x = 0$ , so in a real situation it is usually the starting value.
The gradient is $m = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , and any two points on a straight line give the same value.
A gradient is a rate of change, measured in the units of $y$ per the units of $x$ , such as dollars per month.
A positive gradient means $y$ increases as $x$ increases, a negative gradient means it decreases, a zero gradient means it stays constant, and a vertical line has no gradient.
How steep a line looks depends on the axis scales, so gradients must be compared by calculation, with units.