Problem 70 - A Graphical Approach to College Algebra

Question

Use the analytic method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use your calculator and the standard window to support your conclusion. $$f(x)=x^{4}-5 x+2$$

Step-by-Step Solution

Answer

The function is neither symmetric with respect to the y-axis nor the origin.

Step 1: Check for symmetry with respect to the y-axis

To check for symmetry about the y-axis, substitute \(x\) with \(-x\) in the function and see if the new equation simplifies to the original \(f(x)\). The function is symmetric with respect to the y-axis if \(f(-x) = f(x)\). Calculate \(f(-x)\) for \(f(x)=x^4 - 5x + 2\).\[\begin{align*}\ f(-x) &= (-x)^4 - 5(-x) + 2\ &= x^4 + 5x + 2\end{align*}\] Since \(f(-x)\) is not equal to \(f(x)\), the function is not symmetric about the y-axis.

Step 2: Check for symmetry about the origin

For a function to be symmetric with respect to the origin, \(f(-x)\) must equal \(-f(x)\). We have already determined \(f(-x) = x^4 + 5x + 2\). Now calculate \(-f(x)\).\[\begin{align*}\ -f(x) &= -(x^4 - 5x + 2)\ &= -x^4 + 5x - 2\end{align*}\] Since \(f(-x)\) is not equal to \(-f(x)\), the function is not symmetric about the origin.

Step 3: Conclusion based on symmetry test

The function \(f(x) = x^4 - 5x + 2\) is neither symmetric with respect to the y-axis nor the origin, as neither \(f(-x) = f(x)\) nor \(f(-x) = -f(x)\) holds.

Step 4: Visual Verification with Calculator

Use a graphing calculator to input the function \(f(x) = x^4 - 5x + 2\) and observe the symmetry properties. Viewing the graph confirms no symmetry about the y-axis or the origin as the function looks neither even nor entirely central around the origin.

Key Concepts

Analytic Method

The analytic method is a crucial technique used to determine if a function exhibits symmetry. By performing algebraic manipulations on the function's equation, you can discover inherent symmetries mathematically. To check symmetry using this method, you primarily investigate two types: symmetry about the y-axis and symmetry about the origin. Each has its unique criteria that require substitution and manipulation of the variable within the function. This process allows you to assess a function's characteristics without the immediate need for graphing tools. The key is to substitute the variable or sign to see if certain equality conditions are met in the equations. A detailed exploration of symmetry can reveal a lot about the nature of the function.

Symmetry about the y-axis

Determining if a function is symmetric about the y-axis involves a specific test. The process begins by replacing every instance of the variable \(x\) in the function with \(-x\). The formula \(f(-x) = f(x)\) must hold true for the function to be symmetric about the y-axis. This symmetry is typically seen in even functions; for example, quadratic functions like \(f(x) = x^2\), where squaring the variable makes the sign irrelevant. For our given function \(f(x) = x^4 - 5x + 2\), substituting \(-x\) yields \( f(-x) = x^4 + 5x + 2 \). Since \(f(-x)\) does not equal \(f(x)\), it confirms that the function is not symmetric with respect to the y-axis. This step establishes whether a function maintains its course on a vertical reflection across the y-axis.

Symmetry about the origin

Symmetry about the origin requires the relationship \(f(-x) = -f(x)\). This means that rotating the graph 180 degrees around the origin should leave the function unchanged. Such symmetry is usually present in odd functions, where both axes are mirrored. For our function \(f(x) = x^4 - 5x + 2\), we found that \(f(-x) = x^4 + 5x + 2\) and \(-f(x) = -x^4 + 5x - 2\). Since \(f(-x)\) does not match \(-f(x)\), the function lacks symmetry about the origin. Testing for origin symmetry helps identify if the function has a rotational symmetry characteristic akin to a point reflection.

Graphing Calculator

After applying the analytic method to test for symmetry, a graphing calculator serves as an excellent visual confirmation tool. By inputting the function \(f(x) = x^4 - 5x + 2\) into a graphing calculator using the standard window, you can observe the graph's shape and any symmetry it might possess. With no evident symmetry about the y-axis or the origin, seeing the graph assures that the analytic findings are correct. This visual aid is particularly helpful for students who find abstract algebraic proofs challenging, as it provides immediate, intuitive feedback confirming the absence of such symmetries.