Further Graphs and Tangents

Mathematics revision notes · Cambridge (CIE) · 0580

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1. Recognising the standard shapes

The families you must know

Being able to match an equation to the right picture saves plotting and catches plotting errors
Linear, written y = mx + c, or equivalently as ax + by = c, is a straight line
Quadratic, written y = ax² + bx + c, is a single u-shape or n-shape with one turning point
Reciprocal, written y = a ÷ x or y = a ÷ x + b, is two separate branches that never meet the y-axis, and y = a ÷ x never meets the x-axis either
Cubic, written y = ax³ + b or y = ax³ + bx² + cx, has a long S-like sweep and can turn twice
Exponential, written y = akˣ + b with x in the power, climbs or falls ever more sharply and flattens towards a horizontal line
The sign of the leading number reflects each shape vertically, so a negative cubic falls where a positive one rises

Ten sketches showing each standard shape and its negative twin: y = x a straight line and y = −x its mirror image, y = x² a u-shape and y = −x² an n-shape, y = x³ rising through the origin with y = −x³ falling, y = 1/x with branches in opposite quadrants and y = −1/x with the branches swapped, and y = kˣ climbing to the right against y = k⁻ˣ falling to the right — Source: Types of Graphs by Save My Exams

Fractional powers

Two further shapes appear because this specification allows powers of ½ and −½
The power ½ gives y = √x, a curve that starts at the origin and rises ever more gently
The power −½ gives y = 1 ÷ √x, a curve that drops steeply away from the y-axis and flattens towards the x-axis
Neither exists for negative x, so both curves live entirely to the right of the y-axis

Two sketches on separate axes: y = x to the power one half rising gently from the origin, and y = x to the power minus one half falling steeply from near the y-axis and flattening towards the x-axis. — Source: Types of Graphs by Save My Exams

Worked example

Matching an equation to its shape

Say what shape each of these graphs has: (a) y = 2x + 3 (b) y = x² − 4 (c) y = 6/x (d) y = 3ˣ

Solution:

Read the highest power of x first, because that is what fixes the family
(a) The highest power is 1, so it is a straight line, sloping up with gradient 2
(b) The highest power is 2, so it is a parabola, u-shaped because the number in front of x² is positive
(c) x is on the bottom of a fraction, so it is a reciprocal graph: two separate branches, one in each of two opposite quadrants
(d) x is in the index, not the base, so it is an exponential graph: it never touches the x-axis and climbs ever more steeply
The difference between (b) and (d) is exactly which position x occupies, and reading y = 3ˣ as a cubic is the standard slip

Worked example

Where a cubic meets the axes

(a) Sketch the graph of y = (x − 2)(x + 1)², showing the values where it meets the axes.

(b) Find the exact x-coordinates of the three points where the graph of y = x³ − 6x crosses the x-axis.

Solution:

(a) y = 0 when a factor is zero: x − 2 = 0 gives x = 2, and x + 1 = 0 gives x = −1
The factor (x + 1) is squared, so x = −1 is a repeated root: the curve touches the x-axis there and turns back instead of crossing it
At x = 0, y = (0 − 2) × (0 + 1)² = −2, so the curve cuts the y-axis at (0, −2)
Multiplied out, the first term is +x³, so this is a positive cubic: it rises from the bottom left and leaves at the top right
So the curve climbs to touch the axis at (−1, 0), falls through (0, −2) to a minimum, then rises to cross the axis at (2, 0)
Label −1 and 2 on the x-axis and −2 on the y-axis

A sketch of y = (x − 2)(x + 1)² on plain axes: the curve rises from the bottom left to touch the x-axis at −1 and turn back, falls through −2 on the y-axis to a minimum, then rises to cross the x-axis at 2 and continues up to the top right

(b) Take out the common factor x: x³ − 6x = x(x² − 6) = 0
Either x = 0, or x² = 6, which gives x = √6 or x = −√6
The crossing points have x-coordinates −√6, 0 and √6, left exact as the question asks

2. Completing a table of values

Substituting carefully

A table question gives the equation and a row of x values, and asks for the matching y values
Some entries are already filled in, and those are worth checking against your own working as a test of your method
Bracket every negative x value as you substitute, then apply the order of operations
- For y = 5 − 2x² at x = −3: y = 5 − 2 × (−3)² = 5 − 2 × 9 = −13
A squared term turns a negative input positive, while a cubed term keeps it negative
The calculator's table function will generate the whole row at once from the function, the start value, the end value and the step
A reciprocal term has no value at x = 0, so that cell stays empty rather than being filled with a zero

Worked example

Filling in a row of values

Fill in the missing y values in the table for y = x² − 3x + 1.

x	−1	0	1	2	3	4
y	5			−1

Solution:

At x = 0: 0 − 0 + 1 = 1
At x = 1: 1 − 3 + 1 = −1
At x = 3: 9 − 9 + 1 = 1
At x = 4: 16 − 12 + 1 = 5
Check a given entry the same way, bracketing the negative: at x = −1, (−1)² − 3 × (−1) + 1 = 1 + 3 + 1 = 5, which matches the table
The completed row is 5, 1, −1, −1, 1, 5
The values repeat symmetrically, so the line of symmetry lies half way between x = 1 and x = 2, at x = 1.5

Plotting and drawing

Plot each pair as a cross, accurate to within half a small square
Join the points with a single smooth freehand curve, never with a ruler and never as a series of straight segments
Let the expected shape guide you: a quadratic must be symmetrical, and a reciprocal comes in two separate branches
A point that sits off the pattern is almost always an arithmetic slip, so recheck it before drawing through it
Read the axis scales before plotting, since the two axes rarely use the same one

3. Asymptotes

Lines the curve approaches but never reaches

An asymptote is a straight line that a curve gets closer and closer to without ever touching
Two of them belong to y = a ÷ x: the y-axis, x = 0, and the x-axis, y = 0
- The vertical one comes from the value that would make the bottom zero
- The horizontal one is where the curve settles as x grows very large in either direction
Adding a constant lifts the whole curve, so y = a ÷ x + b has the horizontal asymptote y = b while the vertical one stays at x = 0

The curve y = 1/x with both of its asymptotes drawn as dashed lines and labelled: the horizontal asymptote y = 0 along the x-axis and the vertical asymptote x = 0 along the y-axis, with the two branches approaching each without touching — Source: Types of Graphs by Save My Exams

For positive a, the curve y = a ÷ x² is always positive and both branches sit above the x-axis, unlike y = a ÷ x whose branches sit in opposite quadrants

Two sketches of y = a/x showing what the value of a does: with a far from zero the branches sit well away from the axes, and with a close to zero they are pulled in tight against both axes — Source: Types of Graphs by Save My Exams

Two sketches on separate axes: y = 1 over x with branches in opposite quadrants, and y = 1 over x squared with both branches above the x-axis, each approaching the axes without touching them. — Source: Types of Graphs by Save My Exams

Exponential curves

The curve y = kˣ passes through (0, 1) whatever k is, and has the single asymptote y = 0
It grows when k is greater than 1 and decays towards zero when k lies between 0 and 1
Stretching and shifting gives y = akˣ + b, which meets the y-axis at (0, a + b) and has asymptote y = b
In a growth or decay context, b is the value the quantity settles at in the long run and a + b is its starting value
Asymptotes are written as equations of lines, so the answer is x = 0 or y = 3, never a bare number

Worked example

Writing down the asymptotes

Write down the equations of the asymptotes of y = 6/x, and of y = 2ˣ + 3.

Solution:

An asymptote is a line the curve approaches but never meets, and it is written as an equation, not a number
For y = 6/x, no value of y comes from x = 0, because dividing by zero is undefined
So the curve never crosses the vertical line x = 0, which is the y-axis
However large x becomes, 6/x shrinks towards zero without reaching it
So the curve never meets y = 0, which is the x-axis
For y = 2ˣ + 3, the 2ˣ part shrinks towards zero as x becomes large and negative, but never reaches it
The whole curve therefore stays above 3, getting ever closer to it on the left, giving the asymptote y = 3
The asymptotes are therefore x = 0 and y = 0 for the first curve, and y = 3 for the second, each given as the equation of a line

4. Using a graph to solve an equation

Reading the answers off

The solutions of an equation are the x values where two graphs meet
Where the equation is set equal to zero, the answers are where the curve crosses the x-axis
Where it is set equal to a number, draw that horizontal line and read across
Where two curves are given, the meeting points solve them simultaneously, and there the y values are wanted too

A cubic curve plotted on a grid from x = −3 to 3, crossing the x-axis three times, with a peak near x = −1.3 and a trough near x = 0.8 — Source: Solving equations using graphs by Save My Exams

Read to within half of the smallest square, the accuracy the syllabus sets, and give the x values only, unless the question asks to solve simultaneously

When the equation is not the one plotted

A question often asks you to solve an equation that is not the one drawn
Rearrange the new equation until one side is exactly the expression that was plotted
Whatever is left on the other side is the line you must draw on the grid
Then read off the x values where that line cuts the curve

The same cubic with a straight line drawn across it: the two graphs meet at three points, and dashed arrows drop from each meeting point to the x-axis to give the three solutions — Source: Solving equations using graphs by Save My Exams

Exam tip

Write down the line before you draw it

When you are told to solve an equation by drawing a line on the graph, the line itself earns marks. Write its equation in your working, because that alone often earns one, then rule it right across the grid so it meets the curve at every solution. The answers depend on that line: correct values read off with no correct line drawn usually earn just one special-case mark.

Worked example

Choosing the line to draw

The graph of y = x² − 3x + 1 is drawn on a grid. Explain how to use it to solve x² − 3x − 2 = 0.

Solution:

The plotted expression is x² − 3x + 1, so make that appear on one side of the new equation
Start from x² − 3x − 2 = 0
Add 3 to both sides: x² − 3x + 1 = 3
The left side is now exactly the plotted curve
The line to add to the grid is therefore y = 3
The solutions are the x values where that line meets the curve

The curve y = x² − 3x + 1 on a grid from x = −1 to 4 with the ruled line y = 3 across it; the two meeting points are marked with crosses and dashed lines drop from them to the x-axis at about −0.6 and 3.6

5. Estimating a gradient with a tangent

Drawing the tangent

The gradient of a curve changes from point to point, and the gradient at one point is the gradient of the tangent there
A tangent is a straight line that touches the curve at that point without cutting across it
Rest a ruler against the curve at that point and extend the line right across the grid, so the triangle you read from is large

The curve y = x² − 9x + 8 with a tangent drawn at (1, 0) sloping steeply down and another at (6, −10) sloping up, showing that the gradient of the curve at a point equals the gradient of the tangent there — Source: Finding gradients of tangents by Save My Exams

Read two points far apart on the tangent, ideally on grid intersections, and find the vertical change divided by the horizontal change
- A tangent passing through (1, 2) and (5, 14) has gradient 12 ÷ 4 = 3

A tangent drawn to a falling curve at x = 4, with a right-angled triangle marked on the tangent giving a rise of 2.5 against a run of 4, so the gradient at that point is about −0.6 — Source: Finding gradients of tangents by Save My Exams

Going downhill gives a negative gradient, so put the sign in yourself
The answer is only an estimate, because the tangent is positioned by eye

What the gradient means

The gradient is the rate at which the vertical quantity changes as the horizontal one increases by 1
Its units are the units on the vertical axis divided by the units on the horizontal axis
On a distance–time graph that rate is the speed, and on a speed–time graph it is the acceleration
Answering "what does the gradient represent" needs the quantity named, with its units, not just a number

Exam tip

Leave a ruled tangent on the graph

The tangent itself earns a mark, so rule it on the printed graph, touching the curve at the required point without cutting across it. When the question asks for the gradient of the curve at that point, such as an acceleration, the estimate is credited only alongside a correct tangent or a close attempt at one, so a value with no tangent drawn scores nothing.

Worked example

Reading a tangent off the grid

A tangent drawn to a curve at x = 3 passes through the grid points (1, 9) and (6, −1). Estimate the gradient of the curve at x = 3.

A curve on a grid from x = 0 to 7 that falls to a minimum near x = 5 and rises again, with a ruled tangent touching it at the dot (3, 5) and running down to the right; crosses mark the grid points (1, 9) and (6, −1) on the tangent

Solution:

Vertical change from the first point to the second: −1 − 9 = −10
Horizontal change over the same span: 6 − 1 = 5
Divide the two: −10 ÷ 5 = −2
The gradient of the tangent is −2, and the curve has the same gradient where the tangent touches it
So the gradient of the curve at x = 3 is approximately −2
The tangent slopes downhill, which agrees with the negative answer