Problem 132 - College Algebra

Question

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

Step-by-Step Solution

Answer

To reflect a function's graph about the y-axis, replace every x in the function's equation with -x.

Step 1: Understand What a Reflection Is

A reflection is a transformation that 'flips' a graph across a line, often referred to as the axis of reflection. Reflecting a graph across the y-axis means that the graph will be flipped horizontally, mirroring every point across the y-axis. This changes the position of a point (x, y) on the graph to (-x, y).

Step 2: Relate Reflection to Function's Equation

In terms of a function's equation, reflecting the graph about the y-axis means replacing every x in the equation with -x. This is because for every point (x, y) on the original graph, the reflected point will be (-x, y). Therefore, the value of the function at x in the original equation is the same as the value of the function at -x in the reflected equation.

Step 3: Present the Necessary Change in Equation

Therefore, to reflect a function's graph about the y-axis, you need to replace every x in the function's equation with -x. For example, if the original function is \(f(x) = x^2\), its reflection across the y-axis will be \(f(-x) = (-x)^2\). Notice that for this particular function the equation doesn't change as squaring any real number (positive or negative) yields a positive result, but the concept still applies.