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Chapter 2: Problem 132

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

Short Answer

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

Step by step solution

01

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).
02

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.
03

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.

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Most popular questions from this chapter

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$h(x)=|x-2|+|x+2|$$

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y-f(x-3) $$

If you are given a function's equation, how do you determine if the function is even, odd, or neither?

How can a graphing utility be used to visually determine if two functions are inverses of each other?

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{3}(x-4)$$
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