Problem 44 - College Algebra

Question

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

Step-by-Step Solution

Answer

The x-intercepts are 1 (crosses) and -3 (touches); y-intercept is -9. The graph falls left and rises right.

Step 1: Identify the Intercepts

To find the x-intercepts, set the polynomial equal to zero: \[h(x) = (x-1)^3(x+3)^2 = 0\]. The x-values that satisfy this equation are the x-intercepts: \(x = 1\) and \(x = -3\). The y-intercept is found by evaluating the function at \(x = 0\): \[h(0) = (0-1)^3(0+3)^2 = (-1)^3 \cdot 3^2 = -9\]. The y-intercept is \(y = -9\).

Step 2: Determine the Multiplicity of Each Root

The multiplicity of a root is determined by the power of the factor in the polynomial. For \(x = 1\), the factor is \((x-1)^3\), so the multiplicity is 3. For \(x = -3\), the factor is \((x+3)^2\), so the multiplicity is 2. A root with an odd multiplicity crosses the x-axis, while a root with an even multiplicity touches the x-axis but does not cross it.

Step 3: Analyze End Behavior

End behavior describes the behavior of the polynomial as \(x\) approaches \(\pm \infty\). Since the polynomial \(h(x) = (x-1)^3(x+3)^2\) is of degree 5 (an odd degree), the end behavior is determined by the leading term \(x^5\). As \(x \to -\infty\), \(h(x) \to -\infty\), and as \(x \to \infty\), \(h(x) \to \infty\). This means the graph will fall to the left and rise to the right.

Step 4: Sketch the Graph

Use the intercepts, multiplicity, and end behavior identified in the previous steps to sketch the graph. - The graph intersects the x-axis at \(x = 1\) and \(x = -3\). - At \(x = 1\), the graph crosses the x-axis due to the odd multiplicity of 3. - At \(x = -3\), the graph touches and bounces off the x-axis since the multiplicity is even (2). - The graph falls to the left as \(x\) approaches \(-\infty\) and rises to the right as \(x\) approaches \(\infty\). - Make sure to include the y-intercept at \(y = -9\) in your sketch.

Key Concepts

x-intercepts

When graphing a polynomial, identifying the x-intercepts is a crucial step. The x-intercepts of a polynomial are the points where the graph crosses or touches the x-axis. This occurs when the polynomial equals zero.
For the polynomial function given: \(h(x)=(x-1)^{3}(x+3)^{2}\), to find the x-intercepts, set the polynomial equal to zero: \(h(x) = 0\).

• For \(x-1 = 0\), solving gives you the x-intercept \(x = 1\).
• For \(x+3 = 0\), solving gives you the x-intercept \(x = -3\).

These computations reveal that the graph intersects the x-axis at these points. Each x-intercept's behavior (whether crossing or touching the axis) is further described by its multiplicity.

y-intercept

The y-intercept of a polynomial is where the graph crosses the y-axis. This occurs when the input value \(x = 0\).
To find the y-intercept:

• Plug \(x = 0\) into the polynomial: \(h(0) = (0-1)^3(0+3)^2\).
• Calculate: \((-1)^3 imes 3^2 = -9\).

So, the y-intercept is \(y = -9\). This means that the graph of the polynomial intersects the y-axis at \(y = -9\). It is a single point of reference where the graph passes through the vertical axis.

multiplicity

Multiplicity in a polynomial refers to how many times a particular root (or x-intercept) appears. It is indicated by the exponent of the factor in the polynomial.

• The root \(x = 1\) comes from the factor \( (x-1)^3 \), so its multiplicity is 3.
• The root \(x = -3\) comes from the factor \( (x+3)^2 \), giving it a multiplicity of 2.

Multiplicity affects the behavior of the graph at the x-intercepts.

• If the multiplicity is odd, the graph crosses the x-axis at that root.
• If the multiplicity is even, the graph touches and bounces off the x-axis at that root, rather than crossing it.

In this specific example, at \(x = 1\), the graph crosses due to an odd multiplicity of 3, and at \(x = -3\), it touches and bounces off due to an even multiplicity of 2.

end behavior

The end behavior of a polynomial describes what happens to the value of the polynomial as \(x\) approaches positive or negative infinity. This is largely determined by the leading term, which is the term with the highest degree.

In the polynomial \(h(x)=(x-1)^{3}(x+3)^{2}\), the degree is 5, so the leading term is approximately \(x^5\), indicating:

• As \(x o -\infty\), \(h(x) o -\infty\).
• As \(x o \infty\), \(h(x) o \infty\).

This means that the graph falls to the left and rises to the right, which is typical of odd-degree polynomials where the leading coefficient is positive. Understanding end behavior helps in sketching the overall trajectory of the graph, especially away from the x-intercepts.