The big idea: An asymptote is a straight line that a graph gets infinitely close to but never actually reaches. 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).
What asymptotes look like — y = 1/(x − 3) + 2 has BOTH a vertical asymptote at x = 3 AND a horizontal asymptote at y = 2.
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- 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.
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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
Horizontal asymptote on y = 3·2ˣ + 5 — watch the curve flatten toward the dashed line.
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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 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
Vertical asymptote on y = 1/(x − 3). The curve splits at x = 3 and shoots to ±∞ on either side.
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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 = ...