Problem 16 - Algebra and Trigonometry

Question

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=\frac{1}{5} x(x-5)^{2} $$

Step-by-Step Solution

Answer

The graph passes through (0,0) and (5,0) with a bounce at (5,0). End behavior: falls left, rises right.

Step 1: Identify the Roots

The roots (or zeros) of the polynomial function can be found by setting the equation to zero: \( P(x) = \frac{1}{5} x(x-5)^2 = 0 \). This equation is zero when either \( x = 0 \) or \( (x-5)^2 = 0 \). Solving \( (x-5)^2 = 0 \) gives \( x = 5 \). Thus, the roots are \( x = 0 \) and \( x = 5 \). \( x = 5 \) is a repeated root (multiplicity of 2).

Step 2: Determine the Y-intercept

The y-intercept is found by evaluating the function at \( x = 0 \). Substituting into \( P(x) \): \( P(0) = \frac{1}{5} \times 0 \times (0-5)^2 = 0 \). Therefore, the y-intercept is \( (0,0) \).

Step 3: Analyze the End Behavior

The leading term of \( P(x) \) can be approximated for end behavior by multiplying the highest degrees: \( \frac{1}{5}x^3 \). Since the leading coefficient \( \frac{1}{5} \) is positive and the degree is odd, \( P(x) \) will fall to \(-\infty\) as \( x \rightarrow -\infty \) and rise to \(+\infty\) as \( x \rightarrow +\infty \).

Step 4: Sketch the Graph

Plot the roots \((0,0)\) and \((5,0)\), noting \(x=5\) has a multiplicity of 2, causing the graph to "bounce" at \(x=5\). Consider the y-intercept at \((0,0)\). Take note of the end behavior: as \(x\to-\infty\), \(P(x)\to-\infty\) and as \(x\to+\infty\), \(P(x)\to+\infty\). Begin at the left, pass through each point, showing correct behavior at these critical values, with a bounce at \((5,0)\).

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions helps to visualize the behavior of the function over an extended range of input values. A polynomial function like \( P(x) = \frac{1}{5} x(x-5)^2 \) can be graphed by identifying key characteristics such as roots, y-intercept, and end behavior.

To start, it's essential to plot the roots. In this example, the roots are \( x = 0 \) and \( x = 5 \). These points indicate where the graph will intersect the x-axis. Since \( x = 5 \) is a repeated root (with a multiplicity of 2), the graph will "bounce" off the x-axis at this point.

Next, plot the y-intercept at \( (0,0) \). This point is crucial as it marks where the graph intersects the y-axis. By connecting these intercepts with smooth, continuous lines that reflect the polynomial's behavior, you'll have a foundational sketch of the graph.

Don't forget to consider the end behavior of the function as you complete the sketch to ensure it accurately reflects how the function extends towards infinity.

Roots of Polynomial Functions

The roots, or zeros, of a polynomial function are values of \( x \) where the function equals zero. Finding these roots is an essential part of understanding and graphing the function. For the polynomial \( P(x) = \frac{1}{5} x(x-5)^2 \), the roots are determined by setting the function equal to zero and solving:

\[ P(x) = \frac{1}{5} x(x-5)^2 = 0\]

This equation is satisfied when \( x = 0 \) or \( (x-5)^2 = 0 \). The solution \( x = 0 \) shows a single root, whereas \( x = 5 \) is a repeated root since \( (x-5) \) is squared. A repeated root, in this case, means the graph will touch the x-axis at \( x = 5 \) but not cross it, causing a "bounce."

Understanding the roots' multiplicity helps predict and accurately reflect the graph’s behavior at these critical points on the x-axis.

End Behavior of Polynomial Functions

The end behavior of polynomial functions reveals how the function behaves as \( x \) approaches \( +\infty \) or \( -\infty \). This behavior is mainly determined by the leading term of the polynomial function. For \( P(x) = \frac{1}{5} x(x-5)^2 \), the leading term is \( \frac{1}{5}x^3 \), which is derived from the highest degree terms.

Since the degree of the leading term (3 in this case) is odd and the leading coefficient \( \frac{1}{5} \) is positive, we know that:

• As \( x \to -\infty \), \( P(x) \to -\infty \). The graph falls to the left.
• As \( x \to +\infty \), \( P(x) \to +\infty \). The graph rises to the right.

Understanding the end behavior allows you to draw the graph accurately by showing how it extends beyond the roots and intercepts. It indicates the trajectory that the graph takes as it moves towards the extremes of the coordinate plane.