Problem 85 - Calculus Volume 1

Question

For each polynomial, a. find the degree; b. find the zeros, if any; c. find the \(y\) -intercept(s), if any; d. use the leading coefficient to determine the graph's end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither. \(\quad f(x)=\frac{1}{2} x^{2}-1\)

Step-by-Step Solution

Answer

a) Degree: 2; b) Zeros: \(\pm\sqrt{2}\); c) Y-intercept: \((0, -1)\); d) End behavior: both ends up; e) Even polynomial.

Step 1: Identify the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of \(x\) in \(f(x)=\frac{1}{2} x^{2}-1\) is 2. Thus, the degree of the polynomial is 2.

Step 2: Find the Zeros of the Polynomial

To find the zeros of the polynomial, set \(f(x)\) equal to zero and solve for \(x\).\[\frac{1}{2} x^{2} - 1 = 0\] Add 1 to both sides:\[\frac{1}{2} x^{2} = 1\] Multiply both sides by 2:\[x^{2} = 2\] Take the square root of both sides:\[x = \pm \sqrt{2}\] Therefore, the zeros are \(x = \sqrt{2}\) and \(x = -\sqrt{2}\).

Step 3: Find the Y-intercept

The \(y\)-intercept of a polynomial is the constant term, or \(f(0)\). Substitute \(x = 0\) into the polynomial:\[f(0) = \frac{1}{2}(0)^{2} - 1 = -1\] Therefore, the \(y\)-intercept is \( (0, -1) \).

Step 4: Determine the End Behavior Using the Leading Coefficient

The leading coefficient is \(\frac{1}{2}\), which is positive. Since the degree of the polynomial is even (2), both ends of the graph will point upwards. Thus, as \(x \to \infty, \ f(x) \to \infty\) and as \(x \to -\infty, \ f(x) \to \infty\).

Step 5: Determine if the Polynomial is Even, Odd, or Neither

A polynomial is even if \(f(-x) = f(x)\) and odd if \(f(-x) = -f(x)\). Substitute \(-x\) into \(f(x)\):\[f(-x) = \frac{1}{2} (-x)^{2} - 1 = \frac{1}{2} x^{2} - 1\]This matches \(f(x)\), so the function is even.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept that tells us about the highest power of the variable in the polynomial equation. It determines how many zeros the polynomial can potentially have, the possible shape of its graph, and its end behavior.
For example, in the polynomial function \(f(x) = \frac{1}{2}x^2 - 1\), the highest degree or power of \(x\) is 2. Thus, the degree of this polynomial is 2, which indicates it is a quadratic function. The degree is a key factor in identifying the type of polynomial: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so forth.

Zeros of a Polynomial

Zeros of a polynomial, also known as roots or solutions, are the values of \(x\) for which the polynomial is equal to zero. Finding these zeros gives us important insights into where the graph of the polynomial will intersect the x-axis.
To find the zeros of the polynomial \(f(x) = \frac{1}{2}x^2 - 1\), we set the function equal to zero:

• \(\frac{1}{2}x^2 - 1 = 0\)
• Solving yields \(x = \pm \sqrt{2}\)

These zeros tell us that the polynomial touches the x-axis at \(x = \sqrt{2}\) and \(-\sqrt{2}\). These intersection points are crucial for graphing the polynomial.

Y-Intercept

The y-intercept is the point where the graph of the polynomial crosses the y-axis. It can be easily found by calculating \(f(0)\), substituting 0 for \(x\) in the equation.
In our example, \(f(x) = \frac{1}{2}x^2 - 1\), the y-intercept is found as follows:

\(f(0) = \frac{1}{2}(0)^2 - 1 = -1\)
Therefore, the y-intercept is at point \((0, -1)\). This is a critical point for plotting the function and understanding its position in the coordinate system.

End Behavior of Polynomial Functions

The end behavior of a polynomial is dictated by its degree and the leading coefficient. Specifically, it describes how the polynomial behaves as \(x\) goes to positive or negative infinity.
In the case of \(f(x) = \frac{1}{2}x^2 - 1\):

• The leading coefficient is positive (\(\frac{1}{2}\)).
• The degree is even (2).

These conditions mean that both ends of the graph will converge upwards. As \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\). This helps in plotting the general shape of the graph, even without precise points.

Even and Odd Functions

A polynomial can be categorized as even, odd, or neither by analyzing the symmetry of the function. This classification is based on the results when substituting \(-x\) into the polynomial.
For a function \(f(x)\) to be:

• Even, \(f(-x) = f(x)\).
• Odd, \(f(-x) = -f(x)\).

Taking our function \(f(x) = \frac{1}{2}x^2 - 1\) and substituting \(-x\), we get:

• \(f(-x) = \frac{1}{2}(-x)^2 - 1 = \frac{1}{2}x^2 - 1\)

Since \(f(-x) = f(x)\), \(f(x)\) is an even function. Knowing whether a polynomial is even or odd can provide insight into its symmetry across the y-axis.