Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine whether it is even, odd, or neither. $$f(x)=-x^{2}-8$$

Step-by-Step Solution

Answer

The function \(f(x) = -x^{2}-8\) is an even function.

Step 1: Understand the original function

The function given is \(f(x) = -x^{2}-8\), which is a quadratic function.

Step 2: Check for an even function

A function is even if, when we replace \(x\) with \(-x\), we get back the original function. If we do this for \(f(x)\), we obtain \(-(-x)^{2}-8 = -x^{2}-8\). Here, we see that the result is the same as the original function, suggesting that \(f(x)\) is even. But it's not enough; we need to confirm checking the function graph.

Step 3: Graph the function

By plotting the function on a graph using a graphing utility, we can see whether it's symmetrical about the y-axis (even). After plotting it, we see indeed the graph is symmetrical around the y-axis, this confirms our previous step: the function is even.

Step 4: Conclusion

The function \(f(x) = -x^{2}-8\) is an even function as it is symmetrical about the y-axis, and replacing \(x\) with \(-x\) gets back the original function.

Key Concepts

Even Function

An even function remains unchanged when you substitute \(x\) with \(-x\). This symmetry means that the graph of the function appears identical on both sides of the y-axis. To determine if a function is even, follow these steps:

• Replace each instance of \(x\) in the function with \(-x\).
• Simplify the new expression.
• If the simplified expression matches your original function, you've got an even function.

In our example, the original function is \(f(x) = -x^{2} - 8\). Let's replace \(x\) with \(-x\):\[-(-x)^2 - 8 = -x^2 - 8\].We see that this matches the original function exactly, confirming that our function, \(f(x) = -x^{2} - 8\), is indeed even.

Function Symmetry

Function symmetry plays an essential role in understanding how a function behaves graphically. An even function is one type of symmetrical function.Types of Symmetry:

• Even Symmetry (y-axis symmetry): This is when a function's graph is mirrored over the y-axis. As seen in functions like \(f(x) = -x^2 - 8\), for every point \((x, f(x))\) on the graph, there is a corresponding point \((-x, f(x))\).
• Odd Symmetry (origin symmetry): Different from even functions, these are mirrored over the origin. If you replace \(x\) with \(-x\) and the result is \(-f(x)\), the function is odd.

Understanding these symmetries helps in predicting how functions will look and behave without needing to graph them every time. It can simplify problems and make it easier to catch potential errors in calculations.

Graphing Utility

Graphing utilities are tools, often found in calculators or software, that allow us to visualize functions easily. They're great for observing the behavior of functions and checking for symmetry.With a graphing utility, you can:

• Quickly plot complex functions without manually calculating the points.
• Verify whether functions are symmetrical by observing the plotted graph.
• Identify patterns and behaviors that might not be immediately obvious from the equation alone.

In our example, using a graphing utility to plot \(f(x) = -x^2 - 8\) showed us the symmetry around the y-axis, supporting our algebraic finding that the function is even. These tools are invaluable for visual learners and for confirming analytical results efficiently.