Problem 83 - Calculus Volume 1

Question

For the following exercises, 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. $$ f(x)=2 x^{2}-3 x-5 $$

Step-by-Step Solution

Answer

Degree is 2; Zeros are \(x = \frac{5}{2}\), \(x = -1\); Y-intercept is -5; End behavior: up on both ends; Neither even nor odd.

Step 1: Find the Degree

To find the degree of the polynomial, identify the highest power of the variable (in this case, \(x\)). The polynomial \(f(x) = 2x^2 - 3x - 5\) has a highest power of \(x^2\), so the degree is 2.

Step 2: Find the Zeros

To find the zeros of \(f(x)\), set \(f(x) = 0\) and solve for \(x\). The equation becomes:\[2x^2 - 3x - 5 = 0.\]Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 2\), \(b = -3\), \(c = -5\):\[x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2},\]\[x = \frac{3 \pm \sqrt{9 + 40}}{4},\]\[x = \frac{3 \pm \sqrt{49}}{4},\]\[x = \frac{3 \pm 7}{4}.\]The solutions are \(x = \frac{10}{4} = \frac{5}{2}\) and \(x = \frac{-4}{4} = -1\). Thus, the zeros are \(x = \frac{5}{2}\) and \(x = -1\).

Step 3: Find the Y-intercept

To find the \(y\)-intercept, evaluate \(f(x)\) at \(x = 0\):\[f(0) = 2(0)^2 - 3(0) - 5 = -5.\]Hence, the \(y\)-intercept is \(-5\).

Step 4: Determine End Behavior

The leading coefficient is the coefficient of the term with the highest degree, which is \(2\) in \(2x^2\). Since the leading coefficient is positive and the degree is even, the end behavior for \(f(x)\) is: as \(x\) approaches infinity or negative infinity, \(f(x)\) also approaches infinity.

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

The function is even if \(f(x) = f(-x)\), odd if \(f(-x) = -f(x)\), and neither otherwise. Evaluate:\[f(-x) = 2(-x)^2 - 3(-x) - 5 = 2x^2 + 3x - 5\]Since \(f(x) eq f(-x)\) and \(f(-x) eq -f(x)\), the polynomial is neither even nor odd.

Key Concepts

Degree of the Polynomial

The degree of a polynomial is an important concept because it tells us about the overall behavior of the function. In simple terms, the degree is the highest power of the variable present in the polynomial. For example, in the polynomial function \(f(x) = 2x^2 - 3x - 5\), the term with the highest degree is \(2x^2\). This means that the degree of the polynomial is 2. Understanding the degree helps us in predicting the shape and behavior of the graph. In general, the degree of a polynomial dictates the number of roots and the number of times the graph can cross the x-axis. It's important to note that the degree is always a non-negative whole number.

Zeros of the Polynomial

Zeros, also known as roots, are the x-values where the polynomial equals zero. To find them, we set the polynomial equal to zero and solve for \(x\). For \(f(x) = 2x^2 - 3x - 5\), we solved the equation \(2x^2 - 3x - 5 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 2\), \(b = -3\), and \(c = -5\). Substituting these values gives: \[x = \frac{3 \pm \sqrt{49}}{4}\]This results in two solutions: \(x = \frac{5}{2}\) and \(x = -1\). These zeros are where the graph intersects the x-axis, representing when \(f(x)\) is zero.

Y-Intercept

The y-intercept is a crucial point of the polynomial graph. It's the point where the graph intersects the y-axis. To find it, we substitute 0 for \(x\) and solve for \(f(x)\). For the polynomial \(f(x) = 2x^2 - 3x - 5\), by setting \(x = 0\), we find that: \[ f(0) = 2(0)^2 - 3(0) - 5 = -5 \]Thus, the y-intercept is -5. This point indicates where the graph crosses the y-axis and is useful for sketching the graph and understanding its behavior near the origin.

End Behavior

End behavior describes how the polynomial graph behaves as \(x\) approaches infinity or negative infinity. The leading coefficient and the degree of the polynomial determine this behavior. For \(f(x) = 2x^2 - 3x - 5\), the leading term is \(2x^2\), and it has a positive coefficient with an even degree (2). Therefore, as \(x\) tends toward infinity (either positive or negative), \(f(x)\) also tends toward infinity. In simple terms, both tails of the graph will rise upwards. Understanding the end behavior helps in predicting the shape and limits of the graph as you move away from the center.

Even and Odd Functions

Determining whether a polynomial is even, odd, or neither tells us about its symmetry. An even function meets the condition \(f(x) = f(-x)\), showing symmetry around the y-axis. An odd function meets \(f(-x) = -f(x)\), indicating symmetry about the origin. For \(f(x) = 2x^2 - 3x - 5\), we evaluate: \[ f(-x) = 2(-x)^2 - 3(-x) - 5 = 2x^2 + 3x - 5 \]Because \(f(x) eq f(-x) \) and \(f(-x) eq -f(x)\), the polynomial is neither even nor odd. Understanding these characteristics helps in predicting graph symmetries, aiding in graph sketching.