Function Transformations Visualizer

The parent $f(x)$ is drawn as a dashed teal curve and $g(x)$ as a solid coral curve, on a grid from −10 to 10 on both axes.

The panel above the sliders writes the current equation in terms of f, for example g(x) = −2·f(x − 3) + 1, and lists each transformation as a tag, such as "reflect over x-axis" or "shift right 3". For the horizontal factor, the tag reports $1/|b|$, the actual factor by which horizontal distances change, so b = 2 is labeled a compression by 0.5.

Move one slider at a time first, then combine them and predict where a key point, such as the vertex of $x^2$, will land before you let go. Avoid b = 0: the slider allows it, but then g is constant and the graph collapses to a horizontal line.

Every transformation in AP Precalculus Topic 1.12 is a piece of $$g(x) = a\,f\big(b(x-h)\big) + k.$$ Constants outside f act on outputs (vertical changes). Constants inside f act on inputs (horizontal changes), and they act in the opposite way to what the symbol suggests.

A clean way to track all four at once is the point mapping $$(x,\ y) \ \longmapsto\ \left(\frac{x}{b} + h,\ \ a y + k\right).$$ Apply it to two or three easy points on the parent graph and you have the transformed graph.

Step 1: match the template. With $f(x) = \sqrt{x}$: $a = -2$, $b = 1$, $h = 3$, $k = 1$.

Step 2: name the transformations. Reflect over the x-axis, stretch vertically by a factor of 2, shift right 3, shift up 1.

Step 3: map key points. Using $(x, y) \mapsto (x + 3,\ -2y + 1)$: $(0, 0) \to (3, 1)$, $(1, 1) \to (4, -1)$, $(4, 2) \to (7, -3)$.

Step 4: domain and range. The parent has domain $x \ge 0$ and range $y \ge 0$. After the shift right the domain is $x \ge 3$; after the reflection, stretch and shift up the range is $y \le 1$.

Check it in the simulator: choose √x, set a = −2, h = 3 and k = 1. The equation reads g(x) = −2·f(x − 3) + 1, and the tags read reflect over x-axis, vertical stretch ×2, shift right 3, shift up 1. The coral curve should begin at (3, 1) and pass through (4, −1) and (7, −3).

Follow-up with b. Write $g(x) = (2x - 2)^2$ in the template. Factor the inside first: $2x - 2 = 2(x - 1)$, so $b = 2$ and $h = 1$, not 2. Choose x², set b = 2 and h = 1: the label reads g(x) = f(2(x − 1)), the tags say horizontal compression ×0.5 and shift right 1, and the vertex sits at (1, 0).

Transformations recur throughout the course: Unit 3 sinusoids build amplitude, period, phase shift and midline from these four constants, and exponential and logarithmic models in Unit 2 are often written as transformations of $b^x$ or $\log_b x$. Questions may give a table of values for f and ask for values of g, or show two graphs and ask which transformation links them.