Problem 57 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to decide if the function is odd, even, or neither. $$f(x)=-2 x^{2}+2 x+3$$

Step-by-Step Solution

Answer

The function \(f(x) = -2x^2 + 2x + 3\) is neither even nor odd.

Step 1: Check if the function is even

Plug in \(-x\) for \(x\) in \(f(x)\), simplify the resulting expression and see if it equals to the original function. For the given function, \(f(-x) = -2(-x)^2 + 2(-x) + 3 = -2x^2 - 2x + 3\). This does not equal \(f(x)\) therefore the function is not even.

Step 2: Check if the function is odd

Now, check if the function is odd. This can be done by seeing if negating all terms in \(f(x)\) gives the same result as \(f(-x)\). If we negate all terms in \(f(x)\) we get \(-f(x) = 2x^2 - 2x - 3\). We can see that this also doesn't equal \(f(-x)\), therefore the function is not odd.

Step 3: Graph the function

Finally, to visually confirm our earlier findings, we plot the function using a graphing tool. Upon analyzing the graph, we can see that it's a downward opening parabola. This further verifies that the function is neither even nor odd as it is not symmetric about the y-axis (even function property) or origin (odd function property).

Key Concepts

Even Functions

An even function satisfies the condition \( f(-x) = f(x) \). This means if you substitute \(-x\) into the function and get the original function back, it's even. Even functions have symmetry about the y-axis. When you plot an even function, you’ll notice that the left side is a mirror image of the right side.Some classic examples of even functions include:

• Quadratic functions like \( f(x) = x^2 \)
• Cosine function \( f(x) = \cos(x) \)

In the problem provided, after substituting \(-x\) in the function \( f(x) = -2x^2 + 2x + 3 \), we get \( f(-x) = -2x^2 - 2x + 3 \). Since this is not equal to \( f(x) \), the function is not even. Understanding even functions is vital, as it allows you to identify axes of symmetry and predict the graph's behavior.

Odd Functions

Odd functions meet the condition \( f(-x) = -f(x) \). This means when \(-x\) is substituted in the function, it should return the negative of the original function. Odd functions are symmetric about the origin. This symmetry implies that rotating the graph 180 degrees around the origin yields the same function.Some common examples of odd functions include:

• The cubic function \( f(x) = x^3 \)
• Sine function \( f(x) = \sin(x) \)

For the function in the problem, \( f(-x) = -2x^2 - 2x + 3 \) and \( -f(x) = 2x^2 - 2x - 3 \). Since these expressions are not equal, the function is not odd.Recognizing odd functions is essential, as they often help in simplifying calculations and predicting graph behavior.

Graphing Utilities

Graphing utilities are tools that assist in visually analyzing functions and their properties. They provide an easy visualization of a function’s behavior without manual plotting. These tools are invaluable in confirming whether a function is even, odd, or neither. Advantages of using graphing utilities include:

• Quick and accurate graph creation
• Ability to visualize complex functions easily
• Accessibility for checking symmetry and other properties

For the function in the problem, using a graphing utility helped to visualize the parabola, confirming it neither has y-axis symmetry (even) nor origin symmetry (odd). Graphs act as visual confirmations beyond algebraic tests, ensuring a comprehensive understanding of function behaviors.

Parabolas

A parabola is the graph of a quadratic function and appears as a U-shaped curve. Parabolas can open upwards or downwards, determined by the sign of the leading coefficient.Key features of parabolas include:

• The vertex, which is the highest or lowest point
• The axis of symmetry, a vertical line through the vertex
• The direction of opening, depending on the leading coefficient

The given function, \( f(x) = -2x^2 + 2x + 3 \), is a downward opening parabola due to the negative value of \(-2\) as the leading coefficient.Understanding parabolas is crucial as they frequently appear in various mathematical contexts, from physics to engineering. When we see a parabola's shape on a graph, it immediately informs us about the quadratic nature of the function and its symmetry about the vertex.