What is a function?
A function is a rule that gives exactly one output for each input. We write it as y = f(x): put in an x-value, get out a y-value. Its graph is the set of all points (x, f(x)). Because each input has only one output, any vertical line crosses the graph of a function at most once — the vertical line test.
Parent functions
A parent function is the simplest member of a family:
- Linear, f(x) = x — a straight line through the origin.
- Quadratic, f(x) = x² — a U-shaped parabola with its vertex at the origin.
- Absolute value, f(x) = |x| — a V shape.
- Square root, f(x) = √x — starts at the origin and rises ever more slowly; defined only for x ≥ 0.
- Sine, f(x) = sin x — a wave that repeats every 2π.
- Exponential, f(x) = 2ˣ — doubles with every step to the right and approaches 0 on the left.
- Reciprocal, f(x) = 1/x — two branches separated by asymptotes on both axes.
The four transformations
Every graph in the simulator has the form:
y = a · f( b(x − h) ) + k| Parameter | Effect | Example |
|---|---|---|
| k (outside, added) | Vertical shift: up if k > 0, down if k < 0 | x² + 3 moves up 3 |
| h (inside, subtracted) | Horizontal shift: right if h > 0 | (x − 2)² moves right 2 |
| a (outside, multiplied) | Vertical stretch by |a|; reflection across the x-axis if a < 0 | 3x² is steeper; −x² opens down |
| b (inside, multiplied) | Horizontal compression by |b|; reflection across the y-axis if b < 0 | sin(2x) has half the period |
A useful rule: changes outside the function act on y-values and behave as you expect. Changesinside act on x-values and behave the opposite way — subtracting moves right, and multiplying by 2 squeezes the graph to half its width.
Reading key features
- The y-intercept is f(0), where the graph crosses the y-axis.
- x-intercepts (roots or zeros) are where f(x) = 0.
- For quadratics and absolute value graphs, the vertex is at (h, k).
- For exponential and reciprocal graphs, asymptotes are lines the graph approaches but never reaches.
- The domain is all allowed inputs; the range is all possible outputs.
Key terms
- Transformation
- A change to a graph’s position, size or orientation: translation (shift), stretch/compression, or reflection.
- Translation
- Sliding a graph without changing its shape.
- Period
- The horizontal length of one complete cycle of a repeating function such as sine.
- Asymptote
- A line that a graph approaches more and more closely.
Real-world applications
- Physics: the height of a thrown ball follows a downward-opening parabola; changing launch height is a vertical shift.
- Biology: unlimited population growth and bacterial doubling are exponential.
- Sound and electricity: sine waves describe musical notes and alternating current; amplitude is loudness or voltage and period sets pitch or frequency.
- Chemistry and physics: pressure and volume of a gas at constant temperature follow a reciprocal relationship (Boyle’s law).
- Data science: fitting a transformed function to measurements is the basis of many scientific models.
Common misconceptions
- “(x − 3) shifts left.” It shifts right by 3. Ask: what x makes the inside zero? x = 3.
- “Multiplying by b = 2 stretches the graph wider.” It compresses it horizontally to half the width.
- “The graph touches its asymptote eventually.” For 1/x and 2ˣ, it gets arbitrarily close but never reaches it.
Key takeaways
- y = a·f(b(x − h)) + k: a and k act vertically, b and h act horizontally.
- Outside changes behave as expected; inside changes behave in reverse.
- Negative a reflects across the x-axis; negative b reflects across the y-axis.
- Key features — intercepts, vertex, asymptotes, period — move predictably with the transformations.