Problem 84 - 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. \(f(x)=-3 x^{2}+6 x\)

Step-by-Step Solution

Answer

The degree is 2, zeros are 0 and 2, the y-intercept is 0, both ends go downward, and it is neither even nor odd.

Step 1: Find the Degree

To find the degree of the polynomial, look for the highest power of the variable. Here, the polynomial is \( f(x) = -3x^2 + 6x \). The highest power of \( x \) is 2, so the degree of the polynomial is 2.

Step 2: Find the Zeros

To find the zeros of the polynomial, we solve the equation \( f(x) = 0 \). Set \( -3x^2 + 6x = 0 \). Factor out the common term \( -3x \), giving \( -3x(x - 2) = 0 \). This gives the zeros \( x = 0 \) and \( x = 2 \).

Step 3: Find the Y-intercept

The \( y \)-intercept is the value of the function when \( x = 0 \). Substitute \( x = 0 \) into the polynomial: \( f(0) = -3(0)^2 + 6(0) = 0 \). Therefore, the \( y \)-intercept is 0.

Step 4: Determine End Behavior

The leading coefficient of the polynomial is the coefficient of the term with the highest power of \( x \), in this case, \(-3\). Since the degree is 2 (an even number) and the leading coefficient is negative, both ends of the graph will point downwards as \( x \to -\infty \) and \( x \to \infty \).

Step 5: Determine If Polynomial is Even, Odd, or Neither

A polynomial is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \). Evaluate \( f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x \). Since \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), the polynomial is neither even nor odd.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable present in the expression. In this instance, the polynomial given is \( f(x) = -3x^2 + 6x \).
When examining this polynomial, you will notice that the term with the highest power of \( x \) is \( x^2 \). Therefore, the degree of this polynomial is 2. The degree plays a significant role in defining the properties and behavior, such as the number of zeros it can have and the shape of its graph.

Zeros of a Function

Zeros, also known as roots, of a polynomial, are the values of \( x \) for which the function equals zero. Finding zeros is essential for understanding where the graph of the polynomial will intersect the \( x \)-axis. To find the zeros of our polynomial \( f(x) = -3x^2 + 6x \), set the equation to zero: \(-3x^2 + 6x = 0\).
Factor out the common term \( -3x \) to simplify the expression: \(-3x(x - 2) = 0\).
This gives us two zeros: \( x = 0 \) and \( x = 2 \).
These are the points where the polynomial crosses the \( x \)-axis.

Y-intercept

The \( y \)-intercept of a function is the point at which the graph intersects the \( y \)-axis. It occurs when the value of \( x \) is zero. To find the \( y \)-intercept of the polynomial \( f(x) = -3x^2 + 6x \), substitute \( x = 0 \) into the function: \( f(0) = -3(0)^2 + 6(0) = 0 \).
This means the \( y \)-intercept is at the origin, \( (0, 0) \), which is a key point in graphing and understanding the behavior of the polynomial.

End Behavior

End behavior describes how the graph of a polynomial behaves as the values of \( x \) approach positive or negative infinity. It is influenced by the leading term of the polynomial, which is the term with the highest power of \( x \) and its coefficient. In the polynomial \( f(x) = -3x^2 + 6x \), the leading term is \(-3x^2\).
This polynomial has an even degree (2) and a negative leading coefficient. Because of these properties, both ends of the graph will "point" downwards:

• As \( x \to -\infty \), \( f(x) \to -\infty \)
• As \( x \to \infty \), \( f(x) \to -\infty \)

Understanding end behavior can help predict how the graph will look, especially outside the visible range.

Even and Odd Functions

Determining whether a polynomial is even, odd, or neither helps in predicting symmetries in its graph. For a function \( f(x) \):

• It is even if \( f(-x) = f(x) \). This indicates symmetry about the \( y \)-axis.
• It is odd if \( f(-x) = -f(x) \), indicating rotational symmetry about the origin.

To determine the nature of the given polynomial \( f(x) = -3x^2 + 6x \), calculate \( f(-x) \): \( f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x \).
Notice that \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \).
This means the polynomial is neither even nor odd.