Problem 64 - 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 (0,1) . The \(x\) -intercept is (1,0) . 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 function is \(f(x) = -(x-1)^2x\).

Step 1: Analyze the Given Intercepts

Given the y-intercept at \((0, 1)\), this means that the constant term of the polynomial is 1. The x-intercept at \((1, 0)\) indicates that \((x - 1)\) is a factor of the polynomial.

Step 2: Use Polynomial Degree Information

The degree of the polynomial is 3, indicating it is a cubic polynomial, which can be written as \(f(x) = (x - 1)^a (x - r)(x - s)\) where \(a + 1 = 3\). As there is only one x-intercept given, this suggests a repeated root or a complex conjugate pair.

Step 3: Consider End Behavior

The end behavior \(x \rightarrow -\infty, f(x) \rightarrow \infty\) and \(x \rightarrow \infty, f(x) \rightarrow -\infty\) implies that the leading coefficient is negative. Thus, the polynomial starts high in the left and ends low on the right.

Step 4: Synthesize Polynomial

Considering the factor \((x - 1)\) twice gives \((x - 1)^2\), which still leaves another linear term since we need a cubic. Include another factor \((x + b)\) and set shown intercept conditions: \(f(x) = -(x - 1)^2(x + b)\).

Step 5: Find the Specific Coefficient for Given Points

Substitute the y-intercept \((0, 1)\) into \(f(x) = -(x-1)^2(x+b)\) leading to \[-(-1)^2(b) = 1 \b = -1\]

Step 6: Write the Polynomial Function

Thus the polynomial function is \(f(x) = -(x - 1)^2(x - 0)\), or simplified, \(f(x) = -(x-1)^2x\).

Key Concepts

Intercepts

Intercepts are key points on the graph of a polynomial function. They tell us where the graph crosses the axes.

• The y-intercept occurs where the graph crosses the y-axis, meaning it is the point where x is zero. In this context, the y-intercept is (0, 1), which indicates that the output or constant term of the function is 1.
• The x-intercept happens where the graph crosses the x-axis, meaning it is the point where the output is zero. For this problem, the x-intercept is (1, 0), signifying that when x is 1, the output is zero, so (x - 1) is a factor of the polynomial.

Understanding intercepts helps in constructing the polynomial and knowing where it interacts with the axes simplifies solving and graphing the function dramatically.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It informs us about the overall shape and complexity of the function.

• In our exercise, the degree is provided as 3. This tells us we are dealing with a cubic polynomial, which is a polynomial of degree 3, having the general form expressed as: \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants.
• The degree can also indicate the number of roots, intercepts, and turns in the function. A cubic polynomial might have up to 3 real roots or intercepts and up to 2 turning points.

Knowing the degree is crucial in understanding the basic framework and potential complexity of the graph of the function.

End Behavior

End behavior describes the direction in which the graph of a polynomial moves as the input values become extremely large or small. It is crucial for sketching the behavior of the function at its tails.

• For cubic polynomials, end behavior depends on the leading coefficient and the degree. Here, the end behavior is described by: as \( x \rightarrow -\infty \), \( f(x) \rightarrow \infty \) and as \( x \rightarrow \infty \), \( f(x) \rightarrow -\infty \).
• This behavior suggests a downward opening because, for large values of \( x \), \( f(x) \) decreases. This pattern is characteristic when the leading coefficient is negative in a cubic polynomial.

Understanding end behavior helps predict how the function behaves outside of the key intercept points, allowing for better graphing and comprehension.

Cubic Polynomial

A cubic polynomial is a function involving the third degree of the variable \( x \). It typically takes the form \( f(x) = ax^3 + bx^2 + cx + d \). Being of degree 3 means it is the simplest polynomial that can produce both a turn and an inflection in its graph.

• In our problem, based on given conditions, the polynomial effectively accommodates two (or a repeated) x-intercept following \( (x-1)^2 \), and an additional linear factor to satisfy the degree of 3: resulting in \( f(x) = -(x-1)^2x \).
• This represents a polynomial with 3 parts: a monomial \((x-1)^2 \) recounting a double root, and another \( x \) to achieve the necessary degree.
• Cubic polynomials can model a wide array of behavior and are pivotal in developing understanding of higher degree polynomials.

Thus, constructing the polynomial with a negative leading coefficient and the given intercepts produces this specific cubic function.