E2.11bExtended tier0580 · Section 2: Algebra and graphs

Graphs of functions — 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 touches them); 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 a recent higher-tier exam.

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.

Plotting points inaccurately

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

Seen on a recent higher-tier exam.

Using the rate instead of the multiplier

In $y=ab^x$ , a $15\%$ fall means multiply by $b=0.85$ each step, not $b=0.15$ ; a $4\%$ rise means $b=1.04$ . Using the percentage as $b$ , or subtracting the percentage once instead of compounding, loses marks.

Thinking a reciprocal passes through x = 0

For $y=\dfrac{k}{x}$ there is no value at $x=0$ (division by zero), so the curve never touches the $y$ -axis. Reading a $y$ -value at $x=0$ , or drawing the two branches joined through the origin, is wrong.

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.
Read a starting value straight off
In $y=ab^x$ , $a$ is the value at $x=0$ (since $b^0=1$ ), so read the starting amount straight off. Then substitute a second point to find the multiplier $b$ .
Identify a family by its pattern
Test the points: a constant MULTIPLIER between successive $y$ -values means exponential; a reciprocal is undefined at $x=0$ ; a cubic gives $y=1$ at $x=1$ . Match the pattern before naming the curve.

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

How should you join plotted points on a curve graph?

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Graphs of functions — 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