Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Use the analyric 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 \(a\) calculator and the standand window to support your conclusion. $$f(x)=-x^{3}+2 x$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the origin.

Step 1: Review Symmetry Conditions

To determine symmetry, remember: A function is symmetric with respect to the \(y\)-axis if \(f(-x) = f(x)\), and it is symmetric with respect to the origin if \(f(-x) = -f(x)\). If neither condition holds, the function is neither symmetric.

Step 2: Compute \(f(-x)\)

Substitute \(-x\) into the function \(f(x) = -x^3 + 2x\):\[f(-x) = -(-x)^3 + 2(-x) = x^3 - 2x.\]

Step 3: Check for \(y\)-Axis Symmetry

Compare \(f(-x) = x^3 - 2x\) to \(f(x) = -x^3 + 2x\). They are not equal, so the function is not symmetric with respect to the \(y\)-axis.

Step 4: Check for Origin Symmetry

Compare \(f(-x) = x^3 - 2x\) to \(-f(x) = -(-x^3 + 2x) = x^3 - 2x\). They are equal, indicating the function is symmetric with respect to the origin.

Step 5: Verify with a Graph

Use a calculator to graph \(f(x) = -x^3 + 2x\) and visually check the symmetry against the origin. In a standard window, you should see that reflecting the graph about the origin gives the same graph.

Key Concepts

y-axis symmetry

To check if a function has symmetry about the y-axis, we must determine whether changing the sign of the variable inside the function yields the same function. This checks if each point on the graph has a reflective counterpart directly across the y-axis. Mathematically, this is expressed as

• If \(f(x) = f(-x)\) , then the function is y-axis symmetric.

Let's look at an example: for the function \(f(x) = -x^3 + 2x\), we substitute \(x\) with \(-x\) to find \(f(-x)\). In this case, \(f(-x) = x^3 - 2x\). Comparing this to the original \(f(x) = -x^3 + 2x\), we see that \(f(-x)\) is not equal to \(f(x)\). Thus, this function does not exhibit y-axis symmetry. It's essential to understand that y-axis symmetry is key for even functions, and such functions offer balanced, often simple, mirror-like graphs across the y-axis.

origin symmetry

Origin symmetry in a function means that the graph is unchanged if rotated 180 degrees around the origin. To check for origin symmetry analytically, we ensure that substituting

• \(f(x)\) with \(f(-x)\) yields \(-f(x)\).

For our specific function \(f(x) = -x^3 + 2x\), computing \(f(-x) = x^3 - 2x\) and comparing it with \(-f(x) = x^3 - 2x\), we find they are indeed equal. Therefore, this function has origin symmetry. This kind of symmetry suggests the graph looks the same in all four quadrants if you rotate it around the center point, or origin.Understanding origin symmetry aids us in predicting the behavior of odd functions, which often prominently demonstrate this rotational symmetry.

graphical analysis

Graphical analysis involves looking at a function's graph to verify its properties visually. When testing for symmetry, plotting the function graphically can provide a supportive visual confirmation. Using a calculator can be a robust method for verifying what algebraic methods have revealed.For the function \(f(x) = -x^3 + 2x\), after using symmetry checks analytically, when graphed in a standard calculator window, we should observe that if we rotate the function 180 degrees around the origin, the graph remains unchanged, confirming origin symmetry. Visual analysis can sometimes unveil information missed in algebraic calculations and vice versa. Therefore, it’s a powerful tool paired with algebraic methods to enhance comprehension.

precalculus

Precalculus is a mathematical course that prepares students for calculus, integrating principles from algebra and trigonometry with new concepts that build a foundation for calculus. One such concept is symmetry in functions, a vital visual tool that informs about graph behavior and function characteristics. In precalculus, learning about different types of function symmetry such as y-axis and origin symmetry is crucial. Understanding symmetry not only aids in graphing functions but also facilitates the solving of equations and simplifying expressions. This knowledge is further used for evaluations and predictions in calculus. Precalculus provides students with a scaffold to explore and decode more complex calculus concepts efficiently by honing these fundamental principles.