Cubic graphs: \(y = x^3\)
A cubic is a function whose highest power is \(3\). The simplest is \(y = x^3\). Its graph sweeps up from the bottom left, flattens briefly as it passes through the origin, then climbs steeply to the top right. It has a distinctive "lazy S" shape.
Key features to know:
- It passes through the origin \((0,\ 0)\) for the basic \(y = x^3\).
- It has rotational symmetry about the origin: the bottom-left half is the top-right half turned halfway round.
- A cubic can cross the \(x\)-axis up to three times, and always crosses it at least once, because the two ends head in opposite directions (down on the left, up on the right).
- If the \(x^3\) term is negative, as in \(y = -x^3\), the shape is flipped: it falls from top left to bottom right.
Where this is assessed
Recognising a family from its equation or graph, and naming key features, is Criterion A. Sketching accurately with intercepts and asymptotes labelled is Criterion C. Choosing the right family to model growth or decay is Criterion D.
Exponential graphs: \(y = a^x\)
An exponential function has the variable in the power, like \(y = a^x\) where \(a\) is a positive constant (and \(a \neq 1\)). These curves model fast growth: populations, compound interest, anything that keeps multiplying.
For \(a > 1\) (say \(y = 2^x\)) the graph:
- Always passes through \((0,\ 1)\), because \(a^0 = 1\) for any base.
- Climbs faster and faster to the right, and is always above the \(x\)-axis: \(a^x\) is never zero or negative.
- Flattens towards the \(x\)-axis on the left but never touches it. The line \(y = 0\) is a horizontal asymptote.
Complete a table of values for \(y = 2^x\) at \(x = -1,\ 0,\ 1,\ 2,\ 3\), then state the \(y\)-intercept and the equation of the asymptote.
Reciprocal graphs: \(y = \dfrac{a}{x}\)
A reciprocal function has the variable on the bottom of a fraction, such as \(y = \dfrac{a}{x}\). Its graph comes in two separate curves, one in each of two opposite corners, and it is a classic source of asymptote questions.
- Asymptote
- A straight line that a curve gets closer and closer to but never actually reaches. Reciprocal graphs have two: they approach the axes without touching them.
For \(y = \dfrac{a}{x}\) with \(a > 0\):
- There is a vertical asymptote at \(x = 0\): you can never divide by zero, so the curve never crosses the \(y\)-axis.
- There is a horizontal asymptote at \(y = 0\): as \(x\) grows huge, \(\dfrac{a}{x}\) shrinks towards zero but never gets there.
- The two branches sit in the top-right and bottom-left corners. If \(a\) is negative, they sit in the top-left and bottom-right instead.
- The graph has rotational symmetry about the origin.
Never join the branches
A reciprocal graph is two separate curves. Do not draw a line linking them across the \(y\)-axis, and do not let either branch touch an axis. The asymptotes are walls the curve approaches but never crosses.
Comparing the families
Put side by side, the three families are easy to tell apart. Learn this table and you can name a curve from its equation in seconds.
| Family | Form | Shape | Key features |
|---|---|---|---|
| Cubic | \(y = x^3\) | Lazy S | Through origin; up to 3 roots; no asymptotes |
| Exponential | \(y = a^x\) | Rising curve | Through \((0,1)\); asymptote \(y = 0\); always positive |
| Reciprocal | \(y = \dfrac{a}{x}\) | Two branches | Asymptotes \(x = 0\) and \(y = 0\); no intercepts |
Spot it by the variable
Where is the \(x\)? Raised to a fixed power (\(x^3\)) means a polynomial like a cubic. In the power (\(2^x\)) means exponential. On the bottom of a fraction (\(\tfrac{a}{x}\)) means reciprocal. The position of \(x\) names the family.
Check yourself
Answer from memory of the shapes, then reveal the working.
1. What is the equation of the horizontal asymptote of \(y = 3^x\)? +
An exponential \(y = a^x\) with \(a > 1\) flattens towards the \(x\)-axis on the left but stays positive, so it approaches the line \(y = 0\). That is the horizontal asymptote.
2. For \(y = \dfrac{8}{x}\), find \(y\) when \(x = -2\). +
Substitute \(x = -2\): \(y = \dfrac{8}{-2} = -4\). A negative input gives a negative output, placing the point in the bottom-left branch. So \(y = -4\).
3. How many times can the graph of a cubic cross the \(x\)-axis? +
Because its ends point in opposite directions, a cubic must cross at least once, and it can cross at most three times. So it crosses 1, 2 or 3 times (up to three, at least one).
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