Transformations of Radical Functions

A radical function is a function that contains a variable within a radical (square root, cube root, etc.). The most common form is $$f(x) = \text{sqrt}(x)$$ .

Reflecting a function over the x-axis means that for every point (x, y) on the graph, the point (x, -y) will also be on the graph. This changes the sign of the output.

A vertical stretch occurs when the output values of a function are multiplied by a factor greater than 1, making the graph taller. For example, if $$f(x) = \text{sqrt}(x)$$ is stretched by a factor of 2, it becomes $$g(x) = 2\text{sqrt}(x)$$ .

Shifting a function down by 'k' units involves subtracting 'k' from the function's output. For example, shifting $$f(x) = \text{sqrt}(x)$$ down by 3 gives $$g(x) = \text{sqrt}(x) - 3$$ .

The parent function of a square root function is $$f(x) = \text{sqrt}(x)$$ , which has a domain of [0, ∞) and a range of [0, ∞).

Translating a function right by 'h' units means replacing 'x' with 'x - h' in the function's equation. For example, translating $$f(x) = \text{sqrt}(x)$$ right by 2 gives $$g(x) = \text{sqrt}(x - 2)$$ .

A negative sign in front of a radical function reflects the graph over the x-axis. For example, $$f(x) = -\text{sqrt}(x)$$ is the reflection of $$f(x) = \text{sqrt}(x)$$ .

To identify transformations, compare the function's equation to the parent function. Look for shifts (adding/subtracting inside or outside the radical), stretches (multiplying the radical), and reflections (negative signs).

A vertical stretch by a factor 'a' (where a > 1) makes the graph taller. For example, if $$f(x) = \text{sqrt}(x)$$ is stretched by 3, it becomes $$g(x) = 3\text{sqrt}(x)$$ .

The function is translated right by 3 units and up by 2 units compared to the parent function $$f(x) = \text{sqrt}(x)$$ .