Graph
Gray shows the base f(x); blue shows the transformed y. Dashed lines mark asymptotes and the dot marks the chosen x*.
Result
How it's calculated
Quick guide to y = a·f(b(x−h)) + v
- a scales vertically. If |a| > 1 the graph stretches up/down; if 0 < |a| < 1 it compresses. If a < 0, the graph reflects across the x-axis.
- b scales horizontally. If |b| > 1 the graph compresses (features get closer); if 0 < |b| < 1 it stretches. If b < 0, the graph reflects across the y-axis. (For sine/cosine, period = 2π/|b|.)
- h shifts the graph left/right. Because it is x−h, a positive h shifts the graph to the right.
- v shifts the graph up/down. A positive v shifts the graph up.
- Domain & asymptotes depend on the family. For example, ln(x) needs x > 0 (so ln(x−2) needs x > 2), and 1/x has a vertical asymptote where the inside hits 0.
Worked examples
- Quadratic: pick y = x², a = 2, b = 1, h = 3, v = −1 → y = 2(x−3)² − 1. The vertex moves to (3, −1) and the parabola becomes narrower.
- Sine: pick y = sin x, a = −1, b = 2, h = π/6, v = 1 → the wave flips across the x-axis, the period becomes π, and the whole graph shifts right by π/6 and up by 1.
- Log: pick y = ln x, h = 2 → y = ln(x−2). The vertical asymptote moves to x = 2 and the domain becomes x > 2.
Tip: map points from f(x)
If (t, f(t)) is on the base graph, then it becomes (t/b + h, a·f(t) + v) on the transformed graph (when b ≠ 0). This is a quick way to sanity-check shifts and stretches.
References
FAQ
How do a and b cause reflections?
If b is negative, the graph flips across the y-axis; if a is negative, it flips across the x-axis. The derived table shows Yes/No for each case, and the “How it's calculated” log explains the symmetry in words.
What does the share URL store?
The link stores the family, a, b, h, v, domain preset, x*, and whether teacher mode is on. Anyone opening the link sees the same graph and settings right away.