Problem 73 - 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. $$f(x)=-6$$

Step-by-Step Solution

Answer

The function is symmetric with respect to the y-axis.

Step 1: Understand Symmetry Types

There are two types of symmetry we need to test for: symmetry with respect to the \(y\)-axis and symmetry with respect to the origin. A function \(f(x)\) is symmetric with respect to the \(y\)-axis if \(f(x) = f(-x)\), and symmetric with respect to the origin if \(f(-x) = -f(x)\).

Step 2: Test for y-axis Symmetry

We need to check if \(f(x) = f(-x)\). Substitute \(-x\) into the function: \(f(-x) = -6\). Since \(f(x) = -6\) and \(f(-x) = -6\) are equal, the graph of the function is symmetric with respect to the \(y\)-axis.

Step 3: Test for Origin Symmetry

Check if the function satisfies \(f(-x) = -f(x)\). We have \(f(-x) = -6\) and \(-f(x) = -(-6) = 6\). Since \(f(-x) = -6 eq 6 = -f(x)\), the function is not symmetric with respect to the origin.

Step 4: Use Graphing Calculator

To confirm our conclusion using a calculator, graph the function \(y = -6\). You will see a horizontal line that is symmetric about the \(y\)-axis, confirming the analytic result.

Key Concepts

Graphing Calculator Use

A graphing calculator is an excellent tool for visually confirming the symmetry of a function. You can use it to observe a function's graph and compare it with algebraic findings. Here's how a graphing calculator helps:

• Enter the function in the calculator: Input the given function, in this case, \(f(x) = -6\), in the calculator.
• Observe the graph: When you plot this function, you will notice that the graph is a horizontal line that runs parallel to the x-axis.
• Check the standard window: Typically, the standard window on a graphing calculator will allow you to see a balanced view of what symmetrical relationships the graph might have. For \(f(x) = -6\), the graph is clear and simple.

Using the graphing calculator, we can visually confirm that the graph of the function is symmetric with respect to the \(y\)-axis, supporting the conclusion from our analytic check. The symmetry might not always be as evident without the calculator's help.

Y-Axis Symmetry

Y-axis symmetry in an algebraic function means that the left and right sides of the graph mirror one another across the y-axis. Mathematically, this is determined by testing if \(f(x) = f(-x)\).
For the function \(f(x) = -6\), let's see how this applies. When we substitute \(-x\) into the function, we get \(f(-x) = -6\). Since \(f(x) = -6\) is the same as \(f(-x) = -6\), the equation holds, indicating that there is symmetry about the y-axis.
In essence, this symmetry means that if you fold the graph along the y-axis, the two halves would overlap perfectly. This concept is vital in algebra because it helps simplify analysis by reducing the areas that need examination.

Origin Symmetry

A function shows origin symmetry if its graph looks the same when rotated 180 degrees around the origin. Mathematically, this is expressed as \(f(-x) = -f(x)\).
Let's apply this to \(f(x) = -6\). Substituting \(-x\), we have \(f(-x) = -6\), but \(-f(x)\) equals \(-(-6) = 6\). These expressions are not equal, meaning \(f(-x) eq -f(x)\). Therefore, \(f(x) = -6\) does not have origin symmetry.
Understanding origin symmetry is important because it often signifies a more complex behavior in graphs, requiring careful observation and analysis. It can tell you a lot about how equations react under specific transformations.