Parent Functions
The nine base graphs worth knowing by heart.
The explanation
Rather than memorising hundreds of graphs, learn a handful of originals and how to move them.
The core set:
- Linear: y = x, a straight diagonal
- Quadratic: y = x², a U-shape
- Cubic: y = x³, an S-shape through the origin
- Absolute value: y = |x|, a V
- Square root: y = √x, half a sideways parabola
- Rational: y = 1/x, two curves with the axes as asymptotes
- Exponential: y = 2ˣ, flat then explosive
- Logarithmic: y = log x, the exponential reflected
For each, know its shape, domain, range and whether it passes through the origin. Everything after this unit is one of these shifted, stretched or flipped.
A parent function is the simplest member of a family, with all parameters at their neutral values. Recognising the family determines the domain, range, end behaviour, symmetry and asymptotes before any algebra.
Key facts worth having as reflexes: y = x² and y = |x| are even (symmetric about the y-axis) with range [0, ∞); y = x³ and y = 1/x are odd (symmetric about the origin); y = √x has domain [0, ∞); y = 1/x has domain and range excluding 0, with asymptotes at both axes; y = bˣ has range (0, ∞) and horizontal asymptote y = 0; y = log_b x has domain (0, ∞) and vertical asymptote x = 0.
Exponential and logarithmic functions are inverses, so their graphs are reflections across y = x — which is why one has a horizontal asymptote exactly where the other has a vertical one. The same reflection relates y = x² (restricted to x ≥ 0) and y = √x.
Worked example
Without graphing, state the domain and range of y = √x and y = 1/x.
- √x needs a non-negative radicand: domain [0, ∞).
- Its outputs are the principal roots: range [0, ∞).
- 1/x is undefined at x = 0, and never outputs 0.
Answer: √x: domain and range both [0, ∞). 1/x: domain and range both (−∞,0) ∪ (0,∞).
Common mistakes
- Giving the range of y = x² as all reals.
- Forgetting that 1/x never actually reaches zero.