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Chapter 2: Problem 131

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=x^{2}-1 $$

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

Use a graphing utility to graph the function. Use the graph to determine whether the function has an inverse that is a function (that is, whether the function is one-to-one). $$ f(x)=|x-2| $$

What must be done to a function's equation so that its graph is shrunk horizontally?

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{x-2} $$
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