Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, graph the polynomial functions. Note \(x\) - and \(y\) - intercepts, multiplicity, and end behavior. $$ m(x)=-2 x(x-1)(x+3) $$

Step-by-Step Solution

Answer

The intercepts are \(x = 0, 1, -3\); \(y\)-intercept is 0. The graph falls to the right and rises to the left.

Step 1: Identify the Degree of the Polynomial

The polynomial is given by \(m(x) = -2x(x-1)(x+3)\). Multiply out the terms \[ m(x) = -2(x)(x-1)(x+3) = -2(x^3 + 2x^2 - 3x) \] The degree of the polynomial is 3, as it is the highest exponent of \(x\).

Step 2: Identify End Behavior

For a polynomial of degree 3 with a negative leading coefficient (\(-2\)), the end behavior is as follows:- As \(x \to \infty\), \(m(x) \to -\infty\).- As \(x \to -\infty\), \(m(x) \to \infty\).This is because, for an odd degree polynomial with a negative leading coefficient, the function falls to the right and rises to the left.

Step 3: Find x-intercepts and Their Multiplicities

To find the \(x\)-intercepts, set \(m(x) = 0\): \[-2x(x-1)(x+3) = 0\] Therefore, the \(x\)-intercepts are at \(x = 0, 1, -3\). Each factor appears once, so each intercept has a multiplicity of 1.

Step 4: Find y-intercept

The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the function to find the \(y\)-intercept:\[ m(0) = -2(0)(0 - 1)(0 + 3) = 0 \] So, the \(y\)-intercept is at \((0, 0)\).

Step 5: Graph the Function

Plot the \(x\)-intercepts \((0,0), (1,0), (-3,0)\) on a coordinate plane. Note the \(y\)-intercept is also \((0,0)\). Sketch the curve considering the end behavior. The graph starts from the second quadrant, hits the \(x\)-intercept at \((-3,0)\), rises to a local max, crosses the \(x\)-axis at \((1,0)\), and falls as \(x\) increases beyond this point. Label all intercepts and indicate end behavior on the graph.

Key Concepts

Polynomial Degree

The degree of a polynomial is the highest power of the variable, typically represented by the symbol\( x \), which dictates the shape of the polynomial graph. For the given polynomial \( m(x) = -2x(x-1)(x+3) \), the degree is calculated by multiplying the degrees of each factor:

• The factor \( x \) is of degree 1.
• The factor \( (x-1) \) is also of degree 1.
• The factor \( (x+3) \) carries a degree of 1 too.

When these factors are multiplied together, we find that \( -2x(x-1)(x+3) \) simplifies to a polynomial of degree 3, indicating a cubic function. This means we should expect the graph to have specific characteristics, such as changes in direction, and possibly up to two turning points. The degree helps us understand the basic shape and complexity of the curve.

End Behavior

The end behavior of a polynomial function describes how the function behaves as \( x \) approaches positive or negative infinity. It’s significantly influenced by the degree and the leading coefficient of the polynomial. For the polynomial \( m(x) = -2x(x-1)(x+3) \), having a degree of 3, it's an odd-degree polynomial.Due to its negative leading coefficient (-2), the end behavior can be characterized as follows:

• As \( x \to \infty \), the function \( m(x) \to -\infty \). This means the graph descends on the right side.
• As \( x \to -\infty \), the function \( m(x) \to \infty \). Consequently, the graph ascends on the left side.

This pattern of end behavior is typical for cubic functions with a negative leading coefficient, falling to the right and rising to the left.

X-intercepts

X-intercepts are points where the graph of the polynomial crosses the x-axis, which occur when the output of the function is zero. For the polynomial \( m(x) = -2x(x-1)(x+3) \), the x-intercepts can be found by setting the equation to zero:\[-2x(x-1)(x+3) = 0\]Solving for \( x \) gives:

• \( x = 0 \)
• \( x = 1 \)
• \( x = -3 \)

These solutions clarify that the graph intersects the x-axis at these three points. These intercept points are major indicators of the function's graph, helping to shape our understanding of its trajectory across the x-axis.

Multiplicity

Multiplicity of a root refers to how many times a particular factor occurs in the polynomial. It directly influences the graph's behavior at the intercept corresponding to that root. For \( m(x) = -2x(x-1)(x+3) \):

• Each factor \( x \), \( x-1 \), and \( x+3 \) appears only once.
• Thus, each x-intercept \( x = 0, 1, -3 \) has a multiplicity of 1.

Each x-intercept with a multiplicity of 1 suggests that the graph will cross the x-axis at these points. In polynomial graphs, a root's multiplicity can indicate:

• An odd multiplicity (like 1) means the graph crosses the x-axis at this point.
• An even multiplicity implies the graph touches and rebounds back from the x-axis at this point without crossing it.

Understanding multiplicity is essential as it helps define the graph's interaction with the x-axis, clarifying sections where the graph crosses or just touches the axis.