Families of graphs
What this lesson covers
- Parabolas have roots and a vertex.
- Exponential graphs have a horizontal asymptote.
- Hyperbolas have one vertical and one horizontal asymptote.
The families
- Name the family, then find its features.
| Equation | Shape | Look for |
|---|---|---|
| y = (x − a)(x − b) | Parabola | Roots a and b, vertex halfway |
| y = (x − a)(x − b)(x − c) | Cubic | Three roots |
| y = 2ˣ + c | Exponential | Asymptote y = c |
| y = k/(x − a) + b | Hyperbola | Asymptotes x = a and y = b |
A parabola: y = x² − 4x + 3
- Factorise for the roots, and use symmetry for the vertex.
- y = (x − 1)(x − 3)
- roots at x = 1 and x = 3
- y-intercept = 3put x = 0
- vertex at x = 2, y = −1halfway between the roots
A hyperbola: y = 1/(x − 2) + 3
- The bottom cannot be zero, and the fraction never reaches zero.
- vertical asymptote x = 2
- horizontal asymptote y = 3
- y-intercept: 1/(−2) + 3 = 2.5
Common mistakes
- Drawing a curve that crosses an asymptote of y = 2ˣ + 1.The curve gets close to y = 1 and never reaches it.
- Reading x − 2 as an asymptote at x = −2.Set x − 2 = 0, which gives x = 2.
Your turn
- 1. State the asymptote and y-intercept of y = 2ˣ + 1.
- 2. State the roots and vertex of y = x² − 6x + 8.
Show answer
1. y = 1, and the y-intercept is 2 2. Roots 2 and 4, vertex (3, −1)
What to take away
- Name the family before you draw.
- Every family has its own features to find.
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