GCSE Maths · Algebra

Graphs flashcards

12 cards on Graphs: the key points, worked examples and the slips to watch. Swipe through them, or print them.

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Card 1 of 12

Graphs

Plotting a straight line

How do you plot a straight-line graph from its equation?

Answer it in your head, then turn the card over.

Plotting a straight line

Key points

Make a table: choose some x values and substitute each one to find y.
Plot each (x, y) point, then join them with a ruler.
Work out at least three points: two always make a line, so the third checks your working.

y = 2x − 1

x = −1: y = 2 × (−1) − 1 = −3

x = 0: y = −1, x = 1: y = 1

x = 2: y = 2 × 2 − 1 = 3

Watch out

If one point does not line up with the rest, recheck that substitution: negatives are the usual slip.

Graphs

Reading y = mx + c

In y = mx + c, what do m and c tell you about the line?

Answer it in your head, then turn the card over.

Reading y = mx + c

Key points

m is the gradient: how far the line goes up for every 1 across.
c is the y-intercept: the line crosses the y-axis at (0, c), because x = 0 there.
The gradient is the number in front of x, sign included, wherever that term sits.

Example

y = 3x − 2: gradient 3, crosses the y-axis at (0, −2)

Check: x = 0 gives y = 3 × 0 − 2 = −2

y = 4 − 2x is y = −2x + 4: gradient −2, y-intercept 4

Watch out

In y = 4 − 2x the gradient is −2, not 4: m is whatever multiplies x.

Graphs

Getting y on its own first

How do you find the gradient of a line like 2y = 6x + 4 or 3x + y = 7?

Answer it in your head, then turn the card over.

Getting y on its own first

Key points

Rearrange the equation into the form y = mx + c before reading anything off.
If there is a number in front of y, divide every term by it.
If x is on the same side as y, move it across, changing its sign.

Example

2y = 6x + 4: divide every term by 2

y = 3x + 2, so the gradient is 3 and c = 2

3x + y = 7 becomes y = −3x + 7: gradient −3

Watch out

Reading the gradient of 2y = 6x + 4 as 6 skips the rearranging: it is 3.

Graphs

Gradient from two points

How do you work out the gradient of the line through two points?

Answer it in your head, then turn the card over.

Gradient from two points

Key points

Gradient = change in y ÷ change in x.
Take both changes in the same order: second point minus first point.
Uphill from left to right is positive; downhill is negative.

Formula

gradient = (y₂ − y₁) ÷ (x₂ − x₁)

Through (2, 1) and (4, 9)

Change in y: 9 − 1 = 8

Change in x: 4 − 2 = 2

Gradient = 8 ÷ 2 = 4

Watch out

Change in x ÷ change in y turns the gradient upside down: here it would give $\frac{1}{4}$ instead of 4.

Graphs

The equation of a line from its graph

How do you write down the equation of a straight line drawn on a grid?

Answer it in your head, then turn the card over.

The equation of a line from its graph

Key points

Read c from where the line crosses the y-axis.
Pick two points where the line crosses grid lines exactly, and find the gradient m.
Put both into y = mx + c.
Use the numbers on the axes, not the squares: each square may be worth more than 1.

Through (0, 4) and (2, 0)

c = 4

m = (0 − 4) ÷ (2 − 0) = −4 ÷ 2 = −2

So the line is y = −2x + 4

Watch out

A line that slopes down from left to right always has a negative gradient: check the sign before you finish.

Graphs

Parallel lines

How do you find the equation of a line parallel to a given line through a given point?

Answer it in your head, then turn the card over.

Parallel lines

Key points

Parallel lines have the same gradient, so keep m. Only c changes.
Put the point's x and y into y = mx + c and solve for c.
Lines with the same gradient and different c never meet.

Parallel to y = 2x + 7, through (3, 10)

Same gradient: y = 2x + c

10 = 2 × 3 + c, so c = 10 − 6 = 4

The line is y = 2x + 4

Watch out

Keeping c as well copies the old line: a new point almost always means a new c.

Graphs

Higher

Perpendicular gradients

How do you find the gradient of a line perpendicular to a given line?

Answer it in your head, then turn the card over.

Perpendicular gradients

Key points

The gradients of perpendicular lines multiply to −1.
So take the negative reciprocal: write the gradient as a fraction, flip it, and change its sign.
Check by multiplying the two gradients: you should get −1.

Example

Perpendicular to 2 (that is $\frac{2}{1}$ ): $\frac{- 1}{2}$

Check: 2 × $\frac{- 1}{2}$ = −1

Perpendicular to $\frac{- 3}{4}$ : flip to $\frac{- 4}{3}$ , change the sign: $\frac{4}{3}$

Watch out

Do both steps: perpendicular to 2 is $\frac{- 1}{2}$ , not −2 (sign only) and not $\frac{1}{2}$ (flip only).

Graphs

Higher

The equation of a perpendicular line

How do you find the equation of a line perpendicular to y = 2x + 1 that passes through (4, 3)?

Answer it in your head, then turn the card over.

The equation of a perpendicular line

Key points

Find the perpendicular gradient first: the negative reciprocal of 2 is $\frac{- 1}{2}$ .
Then find c exactly as for a parallel line: put the point into y = mx + c.
Write the answer as y = mx + c.

Example

m = $\frac{- 1}{2}$ , so y = $\frac{- x}{2}$ + c

Put in (4, 3): 3 = $\frac{- 4}{2}$ + c = −2 + c

c = 3 + 2 = 5, so y = $\frac{- x}{2}$ + 5

Watch out

Put the point in the right way round: for (4, 3), x = 4 and y = 3, not the other way round.

Graphs

Distance–time graphs

What do the gradient and a flat section on a distance–time graph tell you?

Answer it in your head, then turn the card over.

Distance–time graphs

Key points

The gradient is the speed: distance ÷ time.
A steeper line means faster. A flat, horizontal section means stopped.
A line sloping back down to the axis is the journey back towards the start.

12 km in the first 2 hours

Speed = gradient = 12 ÷ 2 = 6 km/h

Then flat for half an hour: stopped, speed 0

Watch out

The highest point is the furthest from the start, not the fastest part: look for the steepest section.

Graphs

Conversion graphs

How do you use a conversion graph, and what does a straight line through (0, 0) tell you?

Answer it in your head, then turn the card over.

Conversion graphs

Key points

Find the amount on one axis, go across to the line, then down (or up) to the other axis.
A straight line through (0, 0) means the two amounts are in direct proportion.
So for an amount off the graph, scale up a reading you can make.

5 miles = 8 km; convert 30 miles

30 miles is 30 ÷ 5 = 6 lots of 5 miles

6 × 8 = 48 km

Watch out

Check what one square is worth on each axis before reading anything off.

Graphs

Quadratic graphs

What shape is the graph of a quadratic like y = x² − 4, and how do you plot it?

Answer it in your head, then turn the card over.

Quadratic graphs

Key points

A quadratic (highest power x²) is a symmetrical U shape, called a parabola.
A negative x² term turns it upside down, into an ∩ shape.
Plot points from a table, then join them with a smooth curve, not straight lines.

y = x² − 4

x = −2: y = (−2)² − 4 = 0

x = 0: y = −4, the lowest point

x = 3: y = 3² − 4 = 5, and x = −3 gives 5 too

Watch out

A negative squared is positive: (−3)² = 9, so x = −3 gives the same y as x = 3.

Graphs

Cubic and reciprocal graphs

What do the graphs of y = x³ and y = $\frac{1}{x}$ look like?

Answer it in your head, then turn the card over.

Cubic and reciprocal graphs

Key points

A cubic (highest power x³) runs from bottom left to top right with an S-shaped bend; a negative x³ term flips it.
A reciprocal, y = $\frac{1}{x}$ , is two separate curves in opposite corners.
y = $\frac{1}{x}$ never touches either axis: x cannot be 0, and 1 ÷ x is never 0.

Example

y = x³: x = −2 gives (−2)³ = −8

y = $\frac{1}{x}$ : x = 2 gives 0.5, x = 0.5 gives 2

and x = −1 gives −1, on the other curve

Watch out

A negative number cubed stays negative: (−2)³ is −8, not 8.

Print these cards

Four cards to an A4 sheet, each the size of a standard A6 revision card.

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Each card prints on one face: the question at the top, the notes under it.

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Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.

Put it into practice

Practise GraphsExam-style questions that adapt to how you are doing, with feedback on every answer. Free to start.Learn GraphsEvery syllabus point explained, with worked examples and the video lesson where there is one.

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