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

Step-by-Step Solution

Answer

Degree: 2. Zeros: \(x = 0, 2\). Y-intercept: \((0,0)\). End behavior: both down. Polynomial is odd.

Step 1: Find the Degree of the Polynomial

To determine the degree of a polynomial, identify the term with the highest exponent. In \( f(x) = -3x^2 + 6x \), the term \( -3x^2 \) has the highest exponent of 2. Therefore, the degree of the polynomial is 2.

Step 2: Find the Zeros of the Polynomial

To find the zeros, set the polynomial equal to zero and solve for \(x\): \[-3x^2 + 6x = 0\]Factor the expression:\[-3x(x - 2) = 0\]Set each factor equal to zero:1. \(-3x = 0\) gives \(x = 0\) 2. \(x - 2 = 0\) gives \(x = 2\) So, the zeros are \(x = 0\) and \(x = 2\).

Step 3: Find the Y-Intercept

The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into \(f(x)\): \[f(0) = -3(0)^2 + 6(0) = 0\]Therefore, the \(y\)-intercept is at \((0, 0)\).

Step 4: Determine the Graph's End Behavior

The leading term of the polynomial \(-3x^2\) determines the end behavior. The coefficient \(-3\) is negative, and the degree is even (2), so both ends of the graph will point downward, resembling a downward-facing parabola.

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

To determine if the polynomial is even or odd, check symmetry:- A polynomial is even if \(f(-x) = f(x)\).- A polynomial is odd if \(f(-x) = -f(x)\).Compute \(f(-x)\):\[f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x\]Compare \(f(-x)\) to \(f(x)\):The expression \(f(-x) = -3x^2 - 6x\) is equal to \(-f(x)\), so the polynomial is odd.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a fundamental concept in algebra.It tells us the highest power of the variable within the polynomial, providing insight into the polynomial's behavior and characteristics. For the polynomial function \( f(x) = -3x^2 + 6x \), we identify the term with the highest exponent by examining each term. Here, the term \( -3x^2 \) possesses the highest exponent, which is 2. Thus, the degree of this polynomial is 2. This indicates it is a quadratic polynomial, as polynomials of degree 2 are known.Understanding the degree is crucial, as it helps us predict the shape of the graph and the number of potential zeros, among other properties.

Zeros of a Polynomial

Zeros of a polynomial are the values of \(x\) where the function equals zero.These are essentially the roots, which tell us where the graph intersects the x-axis. To find the zeros, we set the polynomial equal to zero and solve for \(x\). For \( f(x) = -3x^2 + 6x \), set it to zero: \[-3x^2 + 6x = 0\]We can factor the expression as \(-3x(x - 2) = 0\). Setting each factor to zero gives us the solutions \(x = 0\) and \(x = 2\). Therefore, the zeros of the polynomial are at \(x = 0\) and \(x = 2\).Finding the zeros is key to understanding where the graph will touch or cross the x-axis.

Y-Intercept

The y-intercept is where the graph of the polynomial crosses the y-axis.This occurs when \(x\) is 0. For the polynomial \( f(x) = -3x^2 + 6x \), substituting \(x = 0\) gives:\[f(0) = -3(0)^2 + 6(0) = 0\]Thus, the y-intercept is at \((0, 0)\). The y-intercept is an important feature because it tells us the starting point of the graph on the y-axis. It helps in sketching the polynomial's graph and understanding its initial behavior.

End Behavior of Polynomials

The end behavior of a polynomial describes how the graph behaves as \(x\) approaches positive or negative infinity.For the polynomial \( f(x) = -3x^2 + 6x \), the leading term is \(-3x^2\). The sign and degree of this leading term dictate the end behavior.- The degree is 2, an even number.- The leading coefficient is \(-3\), a negative number.Because the degree is even and the coefficient is negative, both ends of the graph will point downwards, much like a downward-facing parabola.This knowledge assists in sketching the graph and predicting its behavior at extreme values of \(x\).

Even and Odd Functions

Determining whether a polynomial is even, odd, or neither requires checking for specific symmetries.- A function is **even** if \(f(-x) = f(x)\) for all \(x\).- A function is **odd** if \(f(-x) = -f(x)\) for all \(x\).For \( f(x) = -3x^2 + 6x \), compute \(f(-x)\):\[f(-x) = -3(-x)^2 + 6(-x) = -3x^2 - 6x\]Noticing that \( f(-x) = -f(x) \), this indicates the polynomial is odd. Recognizing if a function is even or odd can give insights into its graph, showing whether it will have symmetry on the y-axis (even) or origin (odd).