Home / IGCSE Maths 0580 / Algebra and graphs / Graphs of functions: tables, plotting and tangents
Cambridge IGCSE · Topic 2.10
IGCSE Maths 0580 / Topic 2 / IGCSE

From table to curve.

Every graph in Paper 4 starts the same way: a table of values, carefully plotted points and a smooth curve. Get the table right and the rest of the question follows.

Full-screen labGraphs of functions lab → Topic 2All algebra and graphs lessons →
The key idea

Substitute each \(x\) into the equation (brackets round negatives), plot every point, then join them with ONE smooth curve. The gradient of a curve at a point is the gradient of the tangent drawn there.

x y −2−11234 −424 0 run 2 rise 8 y = x² − 2x − 3 tangent at (3, 0) gradient = 8 ÷ 2 = 4
Fig. 1: the table of values for y = x² − 2x − 3 plotted from x = −2 to 4, with the tangent at (3, 0); rise 8 over run 2 gives a gradient of 4
Section 01

Table of values, then plot.

In the lab, pick a function and fill the table one \(x\) at a time. Watch each point land on the grid, then switch on the curve to see why the points must be joined smoothly and not with a ruler.

\(x\)\(-2\)\(-1\)\(0\)\(1\)\(2\)\(3\)\(4\)
\(y = x^2 - 2x - 3\)50\(-3\)\(-4\)\(-3\)05

Practise reading the scale: the graph scales trainer has you work out one small square, plot values and read points back, on printed scales like the exam grids.

Worked example Completing a table for a quadratic
Complete the table for \(y = x^2 - 2x - 3\) at \(x = -2\) and \(x = 1\).
  1. \(x = -2\): \((-2)^2 - 2(-2) - 3 = 4 + 4 - 3\). The square of a negative is positive, and \(-2 \times -2 = +4\).
  2. So \(y = 5\).
  3. \(x = 1\): \(1^2 - 2(1) - 3 = 1 - 2 - 3 = -4\).
  4. Check with symmetry: the values 5, 0, \(-3\), \(-4\), \(-3\), 0, 5 are symmetrical about \(x = 1\), as a quadratic should be.
Answer: \(y = 5\) at \(x = -2\); \(y = -4\) at \(x = 1\)
Worked example A reciprocal graph
Make a table for \(y = \dfrac{6}{x}\) for \(x = -3, -2, -1, 1, 2, 3, 6\) and describe the graph.
  1. Divide 6 by each value: \(-2, -3, -6, 6, 3, 2, 1\).
  2. There is no value at \(x = 0\) because you cannot divide by zero.
  3. Plot the points: they form two separate branches, one in the first quadrant and one in the third.
  4. Each branch gets closer and closer to the axes but never touches them.
Answer: \(y\): \(-2, -3, -6, 6, 3, 2, 1\). Two branches; never join them across \(x = 0\).
Examiner trap

Typing \(-2^2\) into a calculator gives \(-4\), because it squares 2 and then makes it negative. Always type \((-2)^2\) with brackets: the answer is \(+4\). One wrong value ruins the shape of the whole curve.

Section 02

Recognise the shape.

Before you plot, you should already know roughly what the graph will look like. The highest power of \(x\) decides the shape.

EquationNameShape
\(y = ax + b\)linearstraight line, gradient \(a\), meets the \(y\)-axis at \(b\)
\(y = x^2 + ax + b\)quadraticU-shaped parabola with a line of symmetry
\(y = -x^2 + ax + b\)quadraticupside-down parabola (∩ shape)
\(y = \dfrac{a}{x}\)reciprocaltwo branches, never touching the axes
\(y = ax^3 + \dots\) (Extended)cubicS-shaped, can cross the \(x\)-axis up to three times
\(y = ab^x + c\) (Extended)exponentialsteep growth (or decay), levelling towards the line \(y = c\)
Your turn
Name the shape of (a) \(y = 4 - x^2\) (b) \(y = -\dfrac{3}{x}\) (c) \(y = 5 - 2x\).
Examiner trap

Joining the points of a curve with straight ruler lines loses the mark for the curve. Use one smooth freehand curve that passes through every point, and do not flatten the bottom of a parabola between two equal points.

Stage 1 · Learn

Check your understanding

Four quick checks. Each correct answer earns XP and records this skill on your profile.

Quick check+10 XP

For \(y = 2x - 1\), what is \(y\) when \(x = -3\)?

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What shape is the graph of \(y = 4 - x^2\)?

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Which graph never meets either axis?

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(Extended) Where does \(y = 3 \times 2^{x}\) cross the \(y\)-axis?

Section 03

Exponential growth and decay.

Extended. In the lab's growth and decay tab, change the multiplier: above 1 the graph shoots up, between 0 and 1 it falls towards zero. Drag the red point along the curve to read off a value, exactly as you would on Paper 4.

Worked example Growth: reading a time from the graph (Extended)
The number of bacteria after \(t\) hours is \(N = 50 \times 2^{t}\). Make a table for \(t = 0\) to \(4\) and use the graph to estimate when \(N = 300\).
  1. \(t = 0\): \(50 \times 2^0 = 50 \times 1 = 50\). Remember \(2^0 = 1\).
  2. Then \(50 \times 2 = 100\), \(50 \times 4 = 200\), \(50 \times 8 = 400\), \(50 \times 16 = 800\).
  3. Plot \((0, 50), (1, 100), (2, 200), (3, 400), (4, 800)\) and join with a smooth curve.
  4. Draw across from \(N = 300\) to the curve and down to the \(t\)-axis: it meets between 2 and 3, at about 2.6.
Answer: Table: 50, 100, 200, 400, 800. \(N = 300\) when \(t \approx 2.6\) hours.
Worked example Decay: a falling value (Extended)
A car bought for $12 000 loses 20% of its value each year, so \(V = 12\,000 \times 0.8^{t}\). Find its value after 3 years.
  1. Losing 20% leaves 80%, so the multiplier is 0.8 each year.
  2. \(0.8^3 = 0.512\).
  3. \(12\,000 \times 0.512 = 6144\).
Answer: $6144
Examiner trap

Exponential decay is NOT a straight line: after 3 years the car is worth $6144, not \(12\,000 - 3 \times 2400\) = $4800. The 20% is taken from a smaller value each year.

Section 04

Gradient from a tangent.

A curve's steepness changes all the time. To estimate the gradient at one point, rule a tangent: a straight line that just touches the curve at that point without cutting across it there. Then find the gradient of your line using a large triangle.

Worked example Gradient of a quadratic at a point
In Fig. 1 a tangent is drawn to \(y = x^2 - 2x - 3\) at \((3, 0)\). It passes through \((2, -4)\) and \((4, 4)\). Estimate the gradient of the curve at \((3, 0)\).
  1. Rise: \(4 - (-4) = 8\).
  2. Run: \(4 - 2 = 2\).
  3. Gradient \(= \dfrac{\text{rise}}{\text{run}} = \dfrac{8}{2} = 4\).
Answer: Gradient \(\approx 4\)
Worked example A negative gradient
A tangent to \(y = \dfrac{8}{x}\) at \((2, 4)\) meets the \(y\)-axis at \((0, 8)\) and the \(x\)-axis at \((4, 0)\). Find the gradient of the curve at \((2, 4)\).
  1. Going from \((0, 8)\) to \((4, 0)\): the line falls 8 while moving 4 to the right.
  2. Gradient \(= \dfrac{0 - 8}{4 - 0} = \dfrac{-8}{4} = -2\).
  3. The answer is negative because the curve is going downhill at that point.
Answer: Gradient \(= -2\)
Your turn
A tangent to a curve passes through \((1, 2)\) and \((5, 14)\). What is the gradient of the curve where the tangent touches it?
Examiner trap

Use two points on the TANGENT that are far apart, not points on the curve, and read each scale carefully: if the \(y\)-axis goes up in 2s, a rise of 4 squares is 8, not 4.

Stage 2 · Exam

Exam-style questions

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Find the value of \(y = x^2 - 4x + 1\) when \(x = -2\).

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A tangent drawn to a curve at point \(P\) passes through \((1, 2)\) and \((5, 14)\). Estimate the gradient of the curve at \(P\).

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(Extended) The curve \(y = 3 \times 2^{x} + c\) passes through \((0, 5)\). Find \(y\) when \(x = 3\).

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(Extended) A machine costs $15 000 and its value falls by 15% each year. What is its value after 4 years, to the nearest dollar?

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Go deeper · practice
Six original Cambridge-style questions
Tables of values for quadratic, reciprocal and cubic graphs, gradients from tangents, and an exponential growth problem.