IGCSE · Practice questions · Graphs of functions: tables, plotting and tangents

Plot it properly.

Six original Cambridge-style questions on graphs of functions: completing tables, recognising shapes, estimating gradients with tangents and exponential growth.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Table, plot, smooth curve, then read.

Substitute carefully: put negative \(x\) values in brackets, especially when squaring or cubing.
Check the pattern: quadratic tables are symmetrical; a value that breaks the pattern is usually a slip.
Plot and join: one smooth curve through every point; never join reciprocal branches across \(x = 0\).
Tangent: rule a line touching the curve at the point, then gradient = rise ÷ run using a big triangle.
Exponential (Extended): \(ab^x\) starts at \(a\) when \(x = 0\) and is multiplied by \(b\) for each step of 1.

Recall

[3 marks]

Complete the table of values for \(y = x^2 + x - 4\), and state the equation of the line of symmetry of the graph.

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(y\)		\(-2\)		\(-4\)	\(-2\)

\(x = -3\): \(9 - 3 - 4 = 2\). ✓
\(x = -1\): \(1 - 1 - 4 = -4\). ✓
\(x = 2\): \(4 + 2 - 4 = 2\). ✓
Equal \(y\) values at \(x = -1\) and \(x = 0\) (and at \(-2\) and \(1\)), so the line of symmetry is halfway: \(x = -0.5\). ✓

Recall

[2 marks]

(a) Find \(y = \dfrac{8}{x}\) for \(x = -4, -2, -1, 1, 2, 4\).

(b) Explain why the table has no value for \(x = 0\).

(a) \(-2, -4, -8, 8, 4, 2\). ✓
(b) \(\dfrac{8}{0}\) is undefined: you cannot divide by zero, so the graph has no point on the \(y\)-axis. ✓

Calculation

[3 marks]

(Extended) Complete the table of values for \(y = x^3 - 4x + 2\).

\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(y\)		2		2		2

\(x = -3\): \((-3)^3 - 4(-3) + 2 = -27 + 12 + 2 = -13\). ✓
\(x = -1\): \(-1 + 4 + 2 = 5\). ✓
\(x = 1\): \(1 - 4 + 2 = -1\). ✓
\(x = 3\): \(27 - 12 + 2 = 17\). ✓

Analysis

[2 marks]

A tangent is drawn to a curve at the point \(A\). The tangent passes through \((-1, 6)\) and \((2, -3)\).

Estimate the gradient of the curve at \(A\) and say what the sign tells you.

Gradient \(= \dfrac{-3 - 6}{2 - (-1)} = \dfrac{-9}{3} = -3\). ✓
It is negative, so the curve is decreasing (going downhill) at \(A\). ✓

Analysis

[3 marks]

(Extended) The graph of \(y = a \times b^{x}\), where \(b > 0\), passes through \((0, 3)\) and \((2, 48)\).

Find \(a\) and \(b\), and then find \(y\) when \(x = -1\).

At \(x = 0\): \(a \times b^0 = a = 3\). ✓
At \(x = 2\): \(3b^2 = 48\), so \(b^2 = 16\) and \(b = 4\) (positive). ✓
\(x = -1\): \(y = 3 \times 4^{-1} = \dfrac{3}{4} = 0.75\). ✓

Analysis

[4 marks]

(Extended) A lake is stocked with fish. After \(t\) weeks the number of fish is \(P = 200 \times 1.5^{t}\).

(a) Complete a table of values for \(t = 0, 1, 2, 3, 4\).

(b) Draw the graph and use it to estimate when the number of fish reaches 800.

(a) \(200, 300, 450, 675, 1012.5\) (about 1010 fish). ✓
(b) Reading across from \(P = 800\) meets the curve between \(t = 3\) and \(t = 4\), at about \(t = 3.4\) weeks (accept 3.3 to 3.5). ✓
Check: \(200 \times 1.5^{3.4} \approx 794\), close to 800. ✓
(c) The rate at which the number of fish is growing, in fish per week, at that moment. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.