Key features of graphs
Zeros, intercepts and asymptotes are landmarks that help you understand and sketch any graph.
Zeros and x-intercepts
Zeros are the x-values where f(x) = 0. On a graph, they’re the x-intercepts.
| Function | Zeros |
|---|---|
| f(x) = 2x − 6 | x = 3 |
| f(x) = x² − x − 6 = (x − 3)(x + 2) | x = 3, −2 |
| f(x) = x³ − 4x = x(x − 2)(x + 2) | x = 0, ±2 |
| f(x) = 2ˣ | none |
| f(x) = log x | x = 1 |
| f(x) = (x − 5)/(x + 1) | x = 5 |
To find them: set f(x) = 0 and solve.
The y-intercept
Evaluate f(0), if 0 is in the domain.
- f(x) = x² − x − 6: y-intercept −6
- f(x) = 3/(x − 1): y-intercept −3
- f(x) = log x: none (0 not in the domain)
Vertical asymptotes
A vertical asymptote x = a occurs where the function shoots up or down without bound, typically where a denominator is zero (and that factor doesn’t cancel), or at the edge of a log’s domain.
| Function | Vertical asymptote |
|---|---|
| 1/(x − 4) | x = 4 |
| (x + 1)/(x² − 9) | x = 3 and x = −3 |
| log(x + 2) | x = −2 |
| tan x | x = 90°, 270°, … |
Horizontal asymptotes
A horizontal asymptote y = b describes behaviour as x → ±∞.
| Function | Horizontal asymptote |
|---|---|
| 1/x | y = 0 |
| (3x + 1)/(x − 2) | y = 3 |
| 2ˣ + 5 | y = 5 (as x → −∞) |
| (x)/(x² + 1) | y = 0 |
Oblique asymptotes
When the numerator’s degree is exactly one more than the denominator’s, there’s an oblique asymptote (slant line). Divide to find it.
Example: f(x) = (x² + 1)/x = x + 1/x → asymptote y = x.
Holes vs asymptotes
If a factor cancels, there’s a hole instead of an asymptote.
f(x) = (x − 2)(x + 1)/(x − 2) → hole at x = 2 (graph is y = x + 1 with a gap at (2, 3)).
Sketching with landmarks
Example: f(x) = (x − 1)/(x + 2)
- Zero: x = 1
- y-intercept: −½
- Vertical asymptote: x = −2
- Horizontal asymptote: y = 1
- Plot these, then a few test points, and draw two branches.
Worked examples
Example 1: Zeros of f(x) = x² − 16? ±4.
Example 2: Asymptotes of f(x) = 4/(x + 3) − 1? x = −3, y = −1.
Example 3: Oblique asymptote of (x² + 3x + 1)/(x + 1)? Divide: x + 2 − 1/(x + 1) → y = x + 2.
Common mistakes
- Calling every denominator zero an asymptote (check for holes).
- Finding a y-intercept where f(0) is undefined.
- Forgetting zeros of the numerator that cancel.
Try it yourself
- Zeros of f(x) = x(x − 7)?
- y-intercept of f(x) = (x + 6)/(x − 3)?
- Asymptotes of f(x) = (2x)/(x − 5)?
- Is there a hole or an asymptote at x = 4 for (x − 4)/(x² − 16)?
Answers: 1) 0, 7 2) −2 3) x = 5, y = 2 4) a hole (x = −4 is the asymptote)
Quick recap
- Zeros are x-intercepts; f(0) gives the y-intercept.
- Vertical asymptotes come from non-cancelling denominator zeros.
- Horizontal asymptotes and oblique asymptotes describe end behaviour.