The big idea: An asymptote is a straight line that a graph gets infinitely close to but never actually reaches. For AI SL you need to know horizontal asymptotes (the graph levels off) and vertical asymptotes (the graph shoots up or down near a gap in the domain).
- Horizontal asymptote
- A horizontal line y = k that the graph approaches as x → +∞ or x → −∞. The graph flattens towards this line.
- Vertical asymptote
- A vertical line x = a that the graph approaches but never crosses, usually where the function is undefined (e.g. division by zero).
Horizontal asymptote
- Written as y = k
- Graph flattens and runs alongside a horizontal line
- Occurs as x → +∞ or x → −∞
- Example: y = 3·2ˣ + 5 has asymptote y = 5
Vertical asymptote
- Written as x = a
- Graph shoots to ±∞ near a gap in the domain
- Occurs where the function is undefined
- Example: y = 1/(x−3) has asymptote x = 3
How to spot asymptotes on a graph: A horizontal asymptote looks like the curve flattening out and running alongside a horizontal dashed line far to the left or right. A vertical asymptote looks like the graph shooting up or down near a vertical dashed line.
The big idea: A horizontal asymptote tells you the long-run value of a function — what y approaches as x grows very large or very negative. For exponential functions like y = a·bˣ + c, the horizontal asymptote is y = c.
Finding a horizontal asymptote
State the horizontal asymptote of y = 3 · 2ˣ + 5.
Step by step
- Write the general form and identify c.
- As x → −∞, the term 3·2ˣ → 0.
- State the asymptote.
Final answer
y = 5
Write the asymptote as an equation: Always write a horizontal asymptote as y = [value], not just the number. 'The horizontal asymptote is 5' loses marks — write 'y = 5'.
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The big idea: A vertical asymptote at x = a occurs where the function is undefined. The most common case in AI SL is a fraction where the denominator equals zero. The graph shoots towards +∞ or −∞ near that x-value.
Finding a vertical asymptote
Find the vertical asymptote of y = 1 / (x − 3).
Step by step
- Write the condition for undefined: denominator = 0.
- Solve for x.
- State the asymptote.
Final answer
x = 3
Wrong
- Asymptote is 3
- Horizontal asymptote: x = 3
- Vertical asymptote: y = 3
Correct
- Vertical asymptote: x = 3 (write as an equation)
- Horizontal asymptotes use y = ...
- Vertical asymptotes use x = ...
The big idea: End behaviour describes what happens to f(x) as x gets very large (x → +∞) or very negative (x → −∞). For growth models it tells you whether the function grows without bound or levels off towards an asymptote.
| Function type | As x → +∞ | As x → −∞ |
|---|---|---|
| Exponential growth y = a·bˣ (b > 1) | y → +∞ (grows without bound) | y → 0 (levels off to zero) |
| Exponential decay y = a·bˣ (0 < b < 1) | y → 0 (levels off to zero) | y → +∞ (grows without bound) |
| y = a·bˣ + c | Depends on b; shifted up/down by c | Approaches y = c |
| Rational y = k/x | y → 0 | y → 0 |
How to describe end behaviour in words: IB may ask 'describe the behaviour of the function as x increases'. Say: 'As x → +∞, f(x) approaches [value/asymptote] from above/below.' Always reference the asymptote if one exists.