Problem 63 - College Algebra

Question

For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or -1 . There may be more than one correct answer. Th \(y\) -intercept is (0,0) . The \(x\) -intercepts are (0,0) , (2,0) . Degree is 3 . End behavior: as \(x \rightarrow-\infty\), \(f(x) \rightarrow-\infty,\) as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

Step-by-Step Solution

Answer

The polynomial is f(x) = x^3 - 2x^2.

Step 1: Identify Intercepts

The graph passes through the y-intercept at (0, 0) and x-intercepts at (0, 0) and (2, 0). This implies that (x - 0)^2 (x - 2) is a factor of the polynomial. Since the intercept at zero occurs twice, it implies a repeated root at x = 0.

Step 2: Degree and Structure of Polynomial

The degree of the polynomial is 3. Given that a factor of the polynomial is (x - 0)^2 (x - 2) and taking into account the intercept, the polynomial can be represented as f(x) = a(x - 0)^2 (x - 2), where 'a' is the leading coefficient. The polynomial must have one more linear factor.

Step 3: Use End Behavior to Determine Leading Coefficient

End behavior dictates that as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity. This behavior corresponds to a cubic polynomial with a positive leading coefficient. Therefore, the leading coefficient is 1.

Step 4: Formulate the Polynomial Function

Using the known factors and the leading coefficient derived from the end behavior, the polynomial function is f(x) = (x^2)(x - 2). Expand the polynomial to f(x) = x^3 - 2x^2.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is one of its fundamental characteristics. It tells us the highest power of the variable present in the polynomial expression. For example, if you have the polynomial function \(f(x) = x^3 - 2x^2\), the degree is 3. This means the highest exponent of \(x\) is 3.

Understanding the degree of a polynomial is crucial because it gives you insight into the behavior of the graph and the number of roots or solutions the polynomial can have. Here are a few key facts about the degree of polynomials:

• A polynomial of degree \(n\) can have up to \(n\) roots.
• The degree also determines the maximum number of turns the graph of the polynomial can have, which is \(n - 1\).
• The end behavior of the graph is also influenced by the degree, as well as the leading coefficient.

For the polynomial in our exercise, knowing that the degree is 3 tells us the function is a cubic polynomial, which typically has an 'S' shape when plotted due to its three roots and possible turns.

Intercepts of Polynomial Functions

Intercepts are points where the polynomial function crosses the axes on a graph. They are crucial for sketching the overall shape of the graph and understanding where the function has zeros or roots.

• The y-intercept of a polynomial function is the point where the graph crosses the y-axis. This point has coordinates \((0, y)\). For instance, if our polynomial is given by \(f(x) = x^3 - 2x^2\), its y-intercept is at \( (0, 0) \).
• The x-intercepts, conversely, are the points where the graph hits the x-axis, which means the function value is zero at these points. For the same function \(f(x)\), the x-intercepts are \( (0, 0) \) and \( (2, 0) \), showing that these are the roots of the polynomial.

Understanding the intercepts helps in determining the polynomial's factors. In our example, this leads to factors like \((x - 0)^2 (x - 2)\) where \(x - 0\) is a repeated root resulting in \((x - 0)^2\). Thus, intercepts not only tell us about the graph's location but also its algebraic structure.

End Behavior of Polynomials

The end behavior of a polynomial function describes how the function behaves as \(x\) moves towards positive or negative infinity. This is essential in understanding the long-term trends of the polynomial graph.

For any polynomial, the end behavior is determined primarily by two factors:

• The degree of the polynomial
• The sign of the leading coefficient

For a degree 3 polynomial, like \(f(x) = x^3 - 2x^2\), we expect certain end behaviors:

• If the leading coefficient is positive, as \(x\) approaches infinity, \(f(x)\) also goes to infinity. Similarly, as \(x\) heads towards negative infinity, \(f(x)\) goes towards negative infinity.
• If the leading coefficient were negative, it would reverse the behavior, with \(f(x)\) dropping to negative infinity as \(x\) increases, and rising to positive infinity as \(x\) decreases.

In our case, the end behavior (\(f(x)\) going from \(-\infty\) to \(\infty\) as \(x\) goes from left to right) confirms a positive leading coefficient, which matches with our determined polynomial function \(f(x) = x^3 - 2x^2\). Investigating end behavior aids in constructing a visual and conceptual understanding of the function.