10-2 Transformations of Quadratic Functions

Property

Starting from the parent function $f(x) = x^2$, two types of translations shift the parabola without changing its shape:

Vertical shift:

Property

Given the parent quadratic function $f(x) = x^2$, the transformed function is:

Property

For the function $f(x) = x^2$, horizontal transformations are created using $f(ax) = (ax)^2$ where $a > 0$. When $a > 1$, the graph compresses horizontally toward the y-axis by a factor of $\frac{1}{a}$. When $0 < a < 1$, the graph stretches horizontally away from the y-axis by a factor of $\frac{1}{a}$.

Examples