A function assigns exactly one output to each allowed input. In precalculus you study families of functions—linear, quadratic, absolute value, square root, reciprocal, exponential, and more—and you learn to transform their graphs without plotting dozens of points from scratch.
Parent functions
A parent function is the simplest member of a family. Examples:
| Family | Parent | Shape idea |
|---|---|---|
| Linear | y = x | Straight diagonal line |
| Quadratic | y = x² | Parabola opening up |
| Absolute value | y = |x| | V shape |
| Square root | y = √x | Slow rise from the origin |
| Reciprocal | y = 1/x | Two branches; vertical asymptote at x = 0 |
The basic transformations
Start with a parent y = f(x). These rules are used constantly:
| Change | New formula | What happens to the graph |
|---|---|---|
| Vertical shift | y = f(x) + k | Up if k > 0, down if k < 0 |
| Horizontal shift | y = f(x − h) | Right if h > 0, left if h < 0 |
| Vertical stretch | y = a·f(x) | |a| > 1 stretches; 0 < |a| < 1 compresses |
| Reflection | y = −f(x) | Flips over the x-axis |
| Reflection | y = f(−x) | Flips over the y-axis |
A common student trap: f(x − 2) moves the graph right, not left. Think “the inside needs a larger x to feed the same input to f.”
Domain and range
Domain = allowed inputs. Range = possible outputs. Transformations can change both. For example, y = √(x − 3) has domain x ≥ 3 because you cannot take the square root of a negative number in the real numbers. Shifting the parent √x three units right moves the starting point with it.
Why this matters
Word problems rarely hand you a bare x². They hand you “starts at 50, grows, then levels off” or “the lowest point is at hour 3.” Transformations are how you turn that language into a formula and check it against a graph.
Practice: In Hyper-Scale Lab, open Standard → Functions for scenario-based problems with labeled graphs. Start free practice.