Graph
Gray shows the base f(x); blue shows the transformed y. Dashed lines mark asymptotes and the dot marks the chosen x*.
Results
| Horizontal scale 1/|b| | Vertical scale |a| | Reflect over x-axis? | Reflect over y-axis? | Shift (h, v) | Range | Period |
|---|---|---|---|---|---|---|
| 1 | 1 | No | No | (0, 0) | — | — |
How it's calculated
- Model: y = a·f(b(x−h)) + v, f = linear
- Horizontal: x→(x−h)→b(x−h); shift none, scale 1/|b|=1, reflect y-axis? No
- Vertical: y→a·y+v; scale |a|=1, reflect x-axis? No, shift none
- Evaluate x*=0 ⇒ u=b(x*−h)=0, y=a·f(u)+v=0
How to read a, b, h, and v
- a changes vertical scale. If |a| > 1 the graph stretches vertically; if 0 < |a| < 1 it compresses. A negative a reflects across the x-axis.
- b changes horizontal scale. If |b| > 1 the graph compresses horizontally; if 0 < |b| < 1 it stretches. A negative b reflects across the y-axis.
- h controls horizontal shift. Because the form is
x-h, a positive h moves the graph right. A negative h moves it left. - v controls vertical shift. A positive v moves the graph up and a negative v moves it down.
Quick examples
- Quadratic: choose y = x², then set a = 2, h = 3, v = −1 to get a narrower parabola with vertex at (3, −1).
- Sine: choose y = sin x, then set a = −1, b = 2, h = π/6, v = 1 to flip the wave, halve the period, shift right, and move it up.
- Log: choose y = ln x, then set h = 2 to move the vertical asymptote to x = 2 and restrict the domain to x > 2.
Common mistakes
- Reading positive
has a left shift instead of a right shift. - Forgetting that horizontal scaling works in the opposite direction from vertical scaling.
- Changing multiple parameters at once before you understand which one caused the visual change.
- Ignoring family-specific domain limits for log, square-root, and reciprocal graphs.
If (t, f(t)) is on the parent graph, it becomes (t/b + h, a·f(t) + v) on the transformed graph when b ≠ 0. Use that point mapping to sanity-check the graph against the algebra.
FAQ
How do I read transformations of functions from the parent graph?
Start with the parent graph, then change one parameter at a time. h and v shift the graph, while a and b control scaling and reflections.
What does positive h mean in a function transformation?
In y = a·f(b(x−h)) + v, positive h shifts the graph to the right. Negative h shifts it to the left.
How do a and b cause reflections?
If a is negative, the graph reflects across the x-axis. If b is negative, the graph reflects across the y-axis.
Why does |b| > 1 compress horizontally?
Because b multiplies the x-input before f. A feature at x=t on f appears at x=t/b after transformation (plus h), so distances in x become smaller when |b| is larger than 1.
What does the share URL store?
The link stores the family, a, b, h, v, domain preset, x*, and whether teacher mode is on, so the same graph and settings open right away.