3.3cHigher tier4MA1 · Section 3: Sequences, functions and graphs

Cubic, reciprocal and exponential graphs

Key learnings

Know the three shapes
Cubic $y=x^3$ rises through the origin with an S-shape; reciprocal $y=k/x$ is two separate curves with the axes as asymptotes (never touched); exponential $y=k^x$ is always positive and climbs ever faster.
Build a table of values, then join smoothly
Substitute the given x-values to get y, plot the points, then join them with one smooth curve — these graphs are never made of straight segments.
Reciprocal graphs have asymptotes
For $y = k/x$ , $x = 0$ and $y = 0$ are asymptotes: the curve gets closer and closer but never crosses them, and there is no value at $x = 0$ .

Common mistakes

Drawn from real examiner reports.

Joining points with ruled straight lines

Some students used a ruler to draw straight segments between plotted points instead of a smooth curve, particularly around the bottom of the curve.

Seen on November 2024 Paper 1H (Q11).

Plotting points inaccurately

A point such as $(0.5, 5)$ was sometimes plotted inaccurately, distorting the curve and losing the plotting mark.

Seen on November 2024 Paper 1H (Q11).

Sign and power slips in the table

For $y = x^3$ , $(-2)^3 = -8$ (not $-6$ or $+8$ ); for $y = 2^x$ , $2^0 = 1$ (not 0). Errors here ruin the whole curve.

Trying to evaluate a reciprocal at x = 0

For $y = k/x$ there is no value at $x = 0$ (division by zero). Do not plot a point on the y-axis or join the two branches through the origin — they are separate curves.

Multiplying instead of taking a power

For $y = 2^x$ , $2^3$ means $2 \times 2 \times 2 = 8$ , not $2 \times 3 = 6$ . Treating the power as a multiplier wrecks every value in an exponential table.

Exam tips

Use the symmetry/shape as a check
After plotting, check the curve has the expected shape (S-curve, two-branch reciprocal, or always-positive exponential) — an odd kink usually means an arithmetic slip in the table.
Write every value in the table
Show each substituted value before plotting; a visible $2^0 = 1$ or $(-2)^3 = -8$ lets you catch a slip and earns method marks even if one plotted point is slightly off.
Read a solution off with a line
To solve $x^3 = 5$ from the graph, draw the horizontal line $y = 5$ and read the $x$ where it meets the curve, giving the value to the accuracy the scale allows.

Revision notes

Cubic $y = x^3$ : an S-shape through the origin; $(-2)^3 = -8$ , $(2)^3 = 8$ .
Reciprocal $y = \dfrac{k}{x}$ : two separate curves in opposite quadrants. The axes are asymptotes — the curve approaches but never touches $x = 0$ or $y = 0$ , and there is no value at $x = 0$ .
Exponential $y = k^x$ (with $k > 1$ ): always positive, passes through $(0, 1)$ , and rises faster and faster.

Flashcards

Foundation

What shape is y = x³?

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Cubic, reciprocal and exponential graphs

Key learnings

Common mistakes

Exam tips

Revision notes

The three shapes to recognise

Method: table of values → smooth curve

Check the shape

Flashcards

Practice questions