Question

Determine whether the graphs of the following equations and functions are symmetric about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$f(x)=x^{5}-x^{3}-2$$

Short answer

Explain your answer. Answer: The graph of the function \(f(x) = x^5 - x^3 - 2\) is not symmetric about the \(x\)-axis, \(y\)-axis, or the origin. This is because upon testing for each form of symmetry, we found that \(f(x) \neq f(-x)\), \(f(y) \neq -f(y)\), and \(f(x) \neq -f(-x)\), respectively. Additionally, graphing the function confirms the absence of any such symmetry.

Step-by-step solution

Test for symmetry about the \(x\)-axis

To test for symmetry about the \(x\)-axis, we must determine if \(f(x) = f(-x)\). We will do this by replacing \(x\) with \(-x\) in the given function and simplifying: $$f(-x) = (-x)^5 - (-x)^3 - 2 = -x^5 + x^3 - 2$$ Since \(f(x) = x^5 - x^3 - 2\) and \(f(-x) = -x^5 + x^3 - 2\), the function is not symmetric about the \(x\)-axis.

Test for symmetry about the \(y\)-axis

To test for symmetry about the \(y\)-axis, we must determine if \(f(y) = -f(y)\). We will do this by replacing \(x\) with \(y\) in the given function and simplifying: $$f(y) = y^5 - y^3 - 2$$ $$-f(y) = -(y^5 - y^3 - 2) = -y^5 + y^3 + 2$$ Since \(f(y) \neq -f(y)\), the function is not symmetric about the \(y\)-axis.

Test for symmetry about the origin

To test for symmetry about the origin, we must determine if \(f(x) = -f(-x)\). We have already calculated \(f(-x)\) in step 1, so we only need to compare \(f(x)\) and \(-f(-x)\): $$-f(-x) = -(-x^5 + x^3 - 2) = x^5 - x^3 + 2$$ Since \(f(x) = x^5 - x^3 - 2\) and \(-f(-x) = x^5 - x^3 + 2\), the function is not symmetric about the origin.

Graph the function and verify results

Since all symmetry tests resulted in the function not being symmetric, we would not expect to see any form of symmetry when graphing the function \(f(x) = x^5 - x^3 - 2\). You can use graphing software or a graphing calculator to plot the function and confirm that there is no symmetry to the \(x\)-axis, \(y\)-axis, or origin.

Key concepts

Symmetry About the X-axis

Understanding symmetry in graphs helps to visualize and comprehend the behavior of functions. When we explore symmetry about the x-axis, we are looking for a reflection across the x-axis. This means that for any point \( (x, y) \) on the graph, there should be a corresponding point \( (x, -y) \). For a function \( f(x) \) to be symmetric about the x-axis, it must satisfy the condition \( f(x) = - f(x) \) for all values of \( x \).

However, generally, most functions do not exhibit this type of symmetry since it would mean the function outputs both \( y \) and \( -y \) for the same \( x \) value, which goes against the definition of a function. Only certain graphs, like those of even powers of \( x \) in absolute value functions, lie on both sides of the x-axis in a symmetrical fashion.

Symmetry About the Y-axis

In contrast, symmetry about the y-axis is a more common feature in graphs. A graph is symmetrical about the y-axis if for every point \( (x, y) \) there is an equivalent point \( (-x, y) \) reflected across the y-axis. This type of symmetry implies that \( f(x) = f(-x) \) for all real numbers \( x \).

It's a characteristic property of even functions—where the exponents of \( x \) are even numbers. The classic example is the parabola \( f(x) = x^2 \), which opens upward and whose left side is a mirror image of the right side. When testing for this symmetry, we substitute \( -x \) for every instance of \( x \) in the function and simplify; if the original function is retrieved, y-axis symmetry is confirmed.

Symmetry About the Origin

Symmetry about the origin occurs when a graph is rotated 180 degrees around the origin (the point where the x-axis and y-axis intersect) and the graph remains unchanged. For a function to have origin symmetry, each point \( (x, y) \) on the graph requires a corresponding point \( (-x, -y) \). This translates to the algebraic condition \( f(x) = -f(-x) \) for the function \( f(x) \).

Odd functions, where the exponents of \( x \) are odd, typically display this type of symmetry. A simple example of a function with origin symmetry is \( f(x) = x^3 \), which produces a cubic curve that is symmetrical around the origin. During the test for origin symmetry, if reversing the sign of both \( x \) and \( f(x) \) results in the original function, origin symmetry is present.

Function Symmetry Tests

Conducting function symmetry tests allows us to determine if a function has symmetry about the x-axis, y-axis, or origin. These tests serve as a checkpoints:

• For x-axis symmetry: We verify if \( f(x) = - f(x) \).
• For y-axis symmetry: The condition to check is \( f(x) = f(-x) \).
• For origin symmetry: We check whether \( f(x) = -f(-x) \).

By analyzing the function algebraically, these conditions systematically expose the symmetry, if any, inherent in the function's graph. Despite the conclusion of these tests, it is often useful to graph the function for visual confirmation. The graph provides both a visual affirmation of the analytical test and an understanding of the overall shape and behavior of the function.

Graphing Functions

Visual representation is instrumental for understanding functions, and graphing functions is a core skill in mathematics. When graphing, one plots the set of all ordered pairs \( (x, f(x)) \) on a Cartesian coordinate system. Graphs illustrate the properties of functions—like continuity, limits, and, as discussed, symmetry.

The process of graphing involves selecting a range of x-values, calculating the corresponding y-values (using the function rule), and plotting these points. A complete graph emerges from connecting these points smoothly following the behavior defined by the function. With the advancement in technology, graphing calculators and software now assist in creating accurate graphs quickly, which enables more precise symmetry tests and interpretation of the function's behavior.