Reflections and Dilations of Parent Functions

The parent absolute value function is defined as $$f(x) = |x|$$ , which produces a V-shaped graph that opens upwards.

Reflecting a function vertically means to flip the graph over the x-axis, which can be represented by multiplying the function by -1.

To shift a function to the right by 'h' units, you replace 'x' with 'x-h' in the function's equation.

To shift a function down by 'k' units, you subtract 'k' from the function's output, resulting in $$f(x) - k$$ .

A vertical stretch by a factor of 'a' (where a > 1) multiplies the output of the function by 'a', making the graph taller.

The equation is $$y = -|x - 15| - 3$$ .

It shrinks the graph vertically by a factor of 1/3 and translates it up 4 units.

The general form of a quadratic function is $$f(x) = a(x-h)^2 + k$$ , where (h, k) is the vertex.

To apply a horizontal shift of 'h' units to the left, replace 'x' with 'x + h' in the function's equation.

A vertical shift moves the graph up or down without changing its shape; adding 'k' shifts it up, while subtracting 'k' shifts it down.