Consider the relationship between y = f(x) and y = - f(x).

Question

What is this type of function transformation called? What does it look like on a graph?

Answer 1

The transformation from \( y = f(x) \) to \( y = -f(x) \) is called a "reflection" across the x-axis.

Description:

• When you reflect a function across the x-axis, every point \((x, y)\) on the graph of \(y = f(x)\) is transformed to the point \((x, -y)\) on the graph of \(y = -f(x)\).
• This means that for every value of \(y\) in the original function, its corresponding value in the transformed function will have the opposite sign.

Graphical Representation:

• If you have a function \(f(x)\) that takes on values above the x-axis at some points, and below at others, the reflected function \( -f(x) \) will have those values flipped in the vertical direction.
• For instance, if \(f(x)\) has a peak at some point, the graph of \(-f(x)\) will have a trough at the same x-coordinate, and vice versa.

Example:

If \(f(x) = x^2\), then:

• The graph of \(y = f(x)\) is a parabola opening upwards.
• The graph of \(y = -f(x)\) would be a parabola opening downwards, situated symmetrically with respect to the x-axis.

This transformation is visually clear and often used to analyze the behavior of functions when considering their symmetries.