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Chapter 2: Problem 80

\(77-82\) . Test the equation for symmetry. $$ y=x^{2}+|x| $$

Short Answer

Expert verified
The function \(y = x^2 + |x|\) is symmetric about the y-axis (even function).

Step by step solution

01

Identify Types of Symmetry

Functions can exhibit symmetry about the y-axis (even), the origin (odd), or neither. To test for symmetry, we need to compare the original function with its transformations.
02

Test for Y-axis Symmetry (Even Function)

A function is even if substituting \(-x\) results in the original function. Calculate \(y(-x)\): \(y = (-x)^2 + |-x| = x^2 + |x|\).The function doesn鈥檛 change, indicating y-axis symmetry.
03

Test for Origin Symmetry (Odd Function)

A function is odd if substituting \(-x\) results in the negative of the original function, \(-y(x)\). We have \(y(-x) = x^2 + |x|\) and \(-y(x) = -(x^2 + |x|) = -x^2 - |x|\).Since \(y(-x) eq -y(x)\), the function is not symmetric about the origin.
04

Conclusion on Function Symmetry

Since the function satisfies the condition for y-axis symmetry, it is even. There is no origin (odd function) symmetry present in this function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

even function
An even function is one of the basic types of symmetry seen in mathematical functions. For a function to be even, substituting \(x\) with \(-x\) must not change the function. In simpler words, the function should look the same on both sides of the y-axis. This characteristic is also referred to as y-axis symmetry. When you test for y-axis symmetry and \(f(-x) = f(x)\), the function is mathematically defined as even. The function \(y = x^2 + |x|\) is a great example here. Even if you replace every \(x\) with \(-x\), it transforms into \(y = (-x)^2 + |-x| = x^2 + |x|\). Since the function ends up looking identical after the transformation, it proves that the function is even.
  • The graph of an even function will always mirror itself evenly across the y-axis.
  • In terms of practical scenarios, even functions are particularly useful in problems involving balance and symmetry as they inherently divide into equal halves.
odd function
Odd functions display a different type of symmetry known as origin symmetry. To check for an odd function, you try substituting \(x\) with \(-x\) just like an even function, but the outcome should be the negative of the original function, i.e., \(f(-x) = -f(x)\). For the function \(y = x^2 + |x|\), substituting gives us \(y = x^2 + |x|\) for \(y(-x)\), and \(-y(x) = -x^2 - |x|\). This shows \(y(-x) eq -y(x)\), meaning it does not satisfy the condition for being an odd function. Odd functions reflect well on the concept of symmetry about the origin. If you rotate the graph of an odd function 180 degrees around the origin, it will line up perfectly with itself.
  • Graphs of odd functions will have origin symmetry, appearing the same when rotated halfway.
  • Understanding odd functions can help in scenarios involving inverse relationships or alternative directionality in a system.
y-axis symmetry
Y-axis symmetry refers to when the left side of the graph is a mirror image of the right side. This is directly related to even functions, since when a function displays y-axis symmetry, it is even. Take the function \(y = x^2 + |x|\) from the exercise, for example. After substituting \(x\) with \(-x\), we saw the function remained unchanged: \(y = x^2 + |x|\). This result confirmed that the function has y-axis symmetry. Functions that have y-axis symmetry are predictable and balanced, which can be very useful in mathematical modeling and real-world applications.
  • Y-axis symmetry implies that every point on the graph has a corresponding point directly across the y-axis.
  • Recognizing y-axis symmetry makes it easier to draw graphs without plotting numerous points, saving time and effort.

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Most popular questions from this chapter

Kepler's Third Law Kepler's Third Law of planetary motion states that the square of the period \(T\) of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance \(d\) from the sun. (a) Express Kepler's Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about \(2.79 \times 10^{9}\) mi from the sun. Find the period of Neptune.

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(a) Show that if the \(x\) . and \(y\) intercepts of a line are nonzero numbers \(a\) and \(b\) , then the equation of the line can be written in the form $$ \frac{x}{a}+\frac{y}{b}=1 $$ This is called the twe-intercept form of the equation of a line. (b) Use part (a) to find an equation of the line whose \(x\) -intercept is 6 and whose \(y\) -intercept is \(-8\) .

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