Problem 131 - College Algebra

Question

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

Step-by-Step Solution

Answer

To reflect a function's graph about the x-axis, the function's equation must be multiplied by -1. This changes the sign of all y-coordinates, flipping the graph over the x-axis.

Step 1: Understand Reflection Over the X-Axis

To understand what happens when a graph is reflected over the x-axis, consider that each y-coordinate on the graph has its sign changed. Points that were above the x-axis (where y is positive) will end up below the x-axis (where y is negative), and vice versa.

Step 2: Changing the Function's Equation

To reflect a function over the x-axis, the y-coordinates must change sign. This is achieved by taking the values provided by the function and multiplying them by -1. In the equation form, this translates to multiplying the entire function by -1.

Step 3: Applying to a General Function

For a generic function represented by \(y = f(x)\), reflecting the function over the x-axis would result in the new function \(y = -f(x)\). This new function will yield values that are the negative of the original function's values, effectively flipping the graph over the x-axis.