Question

What must be done to a​ function's equation so that its graph is reflected about the​ x-axis?

Answer 1

To reflect a function's graph about the x-axis, negate the entire function, changing the sign of the equation's terms.

Step-by-step explanation:

To understand how a function's graph is reflected about the x-axis, we need to consider the effect of negating the function's equation. Let's denote the original function as f(x), and its equation as y = f(x). When we negate the function, we obtain y = -f(x). This means that for every x-value, the corresponding y-value is multiplied by -1. Consequently, all points on the original graph are reflected across the x-axis.

Mathematically, if the original function is given by
`f(x) = a*x^n + b*x^(n-1) + ... + c`, where 'a,' 'b,' and 'c' are constants, then the reflected function would be -f(x) =
`-a*x^n - b*x^(n-1) - ... - c.` The sign change affects every term in the equation, leading to the reflection. This transformation preserves the shape of the function but changes its orientation. For example, if the original function has a point (2, 3), the reflected function will have a corresponding point at (2, -3) after the reflection about the x-axis.

In summary, reflecting a function about the x-axis involves negating the entire function's equation. This operation is a fundamental transformation in graphing that alters the positioning of points relative to the x-axis, resulting in a mirror image of the original graph.