Problem 45 - College Algebra

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 graph intercepts at (2,0) and (3,0), touching the x-axis at x=2 and crossing at x=3. It decreases as x → -∞ and increases as x → ∞.

Step 1: Identify the Intercepts

To find the x-intercepts, set \(k(x) = 0\): \((x-3)^3(x-2)^2 = 0\). The solutions are \(x = 3\) and \(x = 2\). - At \(x = 3\), the multiplicity is 3 (since the factor \((x-3)\) is raised to the third power).- At \(x = 2\), the multiplicity is 2 (since the factor \((x-2)\) is raised to the second power).For the y-intercept, set \(x = 0\): \(k(0) = (0-3)^3(0-2)^2 = -27 \times 4 = -108\). Therefore, the y-intercept is \((0, -108)\).

Step 2: Analyze Multiplicity

- The factor \((x-3)^3\) implies a root at \(x = 3\) with multiplicity 3, indicating the graph crosses the x-axis at this point with a change of direction (inflection).- The factor \((x-2)^2\) implies a root at \(x = 2\) with multiplicity 2, indicating the graph touches the x-axis and turns around at this point (does not cross).

Step 3: Determine End Behavior

To determine the end behavior, consider the highest degree term. The polynomial can be expanded to its highest degree term as \(x^5\). Since the leading coefficient is positive, as \(x\rightarrow \, \infty, \, k(x) \rightarrow \, \infty\) and as \(x\rightarrow \, -\infty, \, k(x) \rightarrow -\infty\).

Step 4: Sketch the Graph

- Start by plotting the x-intercepts at \((3,0)\) and \((2,0)\), noting the behavior at each point due to its multiplicity.- Plot the y-intercept at \((0, -108)\).- From the end behavior determined, draw the graph starting from the bottom left going upwards towards the first intercept at x=2, touch it and turn back because of the multiplicity 2.- Then, pass through the x-axis at x=3 with a 'flattened' S-shaped crossing due to the multiplicity 3.- Continue drawing upwards, reflecting the positive leading coefficient behavior.

Key Concepts

Intercepts in Polynomial Graphing

When graphing polynomial functions, finding intercepts is crucial. Intercepts are the points where the graph crosses the axes. Let's break it down:

• X-intercepts: These occur where the polynomial function equals zero. For example, in the polynomial \( k(x) = (x-3)^3(x-2)^2 \), set \( k(x) = 0 \) to find the x-intercepts. This solves to \( x = 3 \) and \( x = 2 \). Each corresponds to a point where the graph hits the x-axis, \( (3,0) \) and \( (2,0) \) respectively.
• Y-intercept: This is where the graph crosses the y-axis and can be found by setting \( x = 0 \). For this polynomial, \( k(0) = (0-3)^3(0-2)^2 = -108 \), giving a y-intercept at \( (0, -108) \).

Understanding where these occur helps to establish the basic skeletal structure of the polynomial graph.

Understanding Multiplicity

Multiplicity refers to the number of times a particular root appears. This concept is important for determining how a polynomial graph behaves at its x-intercepts:

• If the multiplicity is odd, like \( (x-3)^3 \), the graph will cross the x-axis but with a flattened, S-shaped curve at \( x = 3 \). This is because an odd multiplicity indicates an inflection or change in direction.
• If the multiplicity is even, such as \( (x-2)^2 \), the graph will touch and turn around at the x-axis without crossing it at \( x = 2 \). An even multiplicity means the graph forms a 'u' or 'n' shape at the intercept.

By analyzing the multiplicity of roots, one can predict the behavior of the polynomial graph around its intercepts.

Identifying End Behavior

End behavior describes how a polynomial graph behaves as \( x \) approaches infinity or negative infinity. This is determined by the polynomial's degree and leading coefficient:

• For the polynomial \( k(x) = (x-3)^3(x-2)^2 \), the term with the highest power is \( x^5 \). This means the graph's overall shape is influenced like an \( x^5 \) function.
• Because the leading coefficient of \( x^5 \) is positive, as \( x \rightarrow +fty \), \( k(x) \rightarrow +fty \). Conversely, as \( x \rightarrow -fty \), \( k(x) \rightarrow -fty \).

This information guides us to sketch the arms of the polynomial graph; starting low from the left and rising high to the right, reflecting the rise and fall expected in such functions.

Graphing Polynomial Functions

Graphing polynomial functions combines all previously discussed elements to depict the function visually:

• Plot the x-intercepts where the roots occur, considering their multiplicity to inform whether the curve crosses or touches and turns at the axis.
• Identify the y-intercept and mark it to give context to the graph's start point.
• Consider the end behavior to sketch how the graph exits to the left and right infinity. For \( k(x) \), start the curve from bottom left, rise through \( x = 3 \), and ascend to top right.

By piecing intercepts, multiplicities, and end behavior together, one can construct a clear, accurate graph of any polynomial function.