Problem 45 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The x-intercepts are at 3 and 2, with end behaviors (+∞, -∞) and y-intercept at -108.

Step 1: Identify the x-intercepts

The x-intercepts occur where the function equals zero. For \(k(x)=(x-3)^3(x-2)^2\), set each factor equal to zero: \(x-3=0\) and \(x-2=0\). Thus, the x-intercepts are \(x=3\) and \(x=2\).

Step 2: Determine the multiplicity of each root

The multiplicity of a root is the exponent of the factor. For \((x-3)^3\), the root \(x=3\) has a multiplicity of 3. For \((x-2)^2\), the root \(x=2\) has a multiplicity of 2.

Step 3: Determine the y-intercept

The y-intercept occurs where \(x=0\). Substitute \(x=0\) into the function: \(k(0)=(0-3)^3(0-2)^2=(-3)^3(-2)^2=-27 \times 4=-108\). Thus, the y-intercept is at \(y=-108\).

Step 4: Analyze end behavior

The end behavior is determined by the leading term when the polynomial is expanded. The highest degree term is \(x^5\), which is positive, so as \(x\to\infty\), \(k(x)\to\infty\), and as \(x\to -\infty\), \(k(x)\to -\infty\).

Step 5: Sketch the graph using key features

Plot the x-intercepts \(x=3\) and \(x=2\), noting that \(x=3\) has odd multiplicity (graph passes through) and \(x=2\) has even multiplicity (graph touches at the axis). Plot the y-intercept at \(y=-108\) and draw the end behavior as described in Step 4.

Key Concepts

Understanding x-intercepts

In polynomial functions, x-intercepts are the points where the graph crosses or touches the x-axis. These intercepts are found by setting the polynomial equation equal to zero and solving for the values of x. For the polynomial function \(k(x)=(x-3)^3(x-2)^2\), we find the x-intercepts by solving the equations \(x-3=0\) and \(x-2=0\). This gives us the x-intercepts at \(x=3\) and \(x=2\).

• At \(x=3\): The graph will either cross or touch the x-axis.
• At \(x=2\): Similarly, the graph will interact with the x-axis.

Finding x-intercepts is like identifying the points where the graph meets the horizontal ground. It's a key step in sketching the graph of a polynomial.

What is multiplicity?

Multiplicity refers to the number of times a particular x-intercept is repeated. It is indicated by the exponent of the factor in the polynomial. In \(k(x)=(x-3)^3(x-2)^2\), each factor's exponent tells us the multiplicity:

• \(x=3\) has multiplicity 3: This means the graph will cross the x-axis at \(x=3\), creating a visible curve due to the odd multiplicity.
• \(x=2\) has multiplicity 2: As this is an even multiplicity, the graph will "bounce" off the x-axis at \(x=2\), touching but not crossing.

The concept of multiplicity helps understand how the graph behaves at each x-intercept, providing insight into whether the graph will simply touch or actually cross the x-axis.

Finding the y-intercept

The y-intercept occurs where the graph crosses the y-axis, or when \(x=0\). To find the y-intercept of \(k(x)\), we substitute \(x=0\) in the function: \[(0-3)^3(0-2)^2 = (-3)^3(-2)^2 = -27 \times 4 = -108.\]
Thus, the y-intercept is at \(y=-108\).

• The y-intercept is a useful point to plot, helping anchor the graph vertically.
• It shows the function's value when all terms linked with x are at their simplest form.

The y-intercept provides an important fixed point about which the overall form of the polynomial graph is constructed.

Examining end behavior

The end behavior of a polynomial function describes how the graph behaves as \(x\) approaches positive or negative infinity. For \(k(x)=(x-3)^3(x-2)^2\), the leading term, which is derived from multiplying all the highest degree terms of each factor, is \(x^5\).

• The degree (5) being odd and the leading coefficient being positive means the graph will rise to infinity as \(x\) approaches positive infinity.
• Conversely, as \(x\) approaches negative infinity, the graph will fall to negative infinity.

Understanding end behavior gives a wide perspective on how the function behaves in larger scopes, ensuring the sketch reflects accurate trends at the edges of the graph.