Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{3}(x-2) $$

Step-by-Step Solution

Answer

The intercepts are \((0,0)\) and \((2,0)\). Both ends rise.

Step 1: Factor the Polynomial

The given polynomial function is already factored: \[ f(x) = x^3 (x - 2) \]. This will help us easily determine the roots and analyze the graph.

Step 2: Determine the Intercepts

To find the **x-intercepts**, set the polynomial equation to zero:\[ x^3 (x - 2) = 0 \].This gives us the intercepts at \(x = 0\) (with multiplicity 3) and \(x = 2\). The **y-intercept** occurs when \(x = 0\), hence the point is \((0, 0)\).

Step 3: Use a Calculator for Graph

Using a graphing calculator, plot the function \(f(x) = x^3 (x - 2)\). Observe the overall shape and direction of the graph.

Step 4: Analyze End Behavior

Identify how the function behaves as \(x\) approaches infinity and negative infinity.Due to the leading term \(x^4\), as \(x \to \infty\), \(f(x) \to \infty\) and as \(x \to -\infty\), \(f(x) \to \infty\).

Step 5: Create a Table for Confirmation

Make a table with values of \(x\) and corresponding \(f(x)\) to confirm the end behavior:\[\begin{array}{|c|c|}\hline x & f(x) \\hline -3 & 81 \ -2 & -32 \ -1 & -3 \ 0 & 0 \ 1 & 1 \ 2 & 0 \ 3 & 81 \\hline\end{array}\]The table supports that as \(x\) approaches large positive or negative values, \(f(x)\) values are increasingly positive.

Key Concepts

Intercepts in Graphs

Intercepts are crucial in understanding the behavior and position of a polynomial function on a graph. They provide us with information on where the function intersects the x-axis and the y-axis.

• X-intercepts: These are the points where the graph crosses the x-axis. To find them for the polynomial \(f(x) = x^3(x-2)\), we set the polynomial equal to zero: \(x^3(x-2) = 0\). Solving for \(x\) gives the roots \(x = 0\) and \(x = 2\). The intercept \(x = 0\) has a multiplicity of 3, suggesting the graph is tangent to the x-axis at this point.
• Y-intercept: This is where the graph crosses the y-axis, found by evaluating the function at \(x = 0\). For \(f(0)\), we substitute zero into the equation, resulting in \(f(0) = 0\). Thus, the graph intersects the y-axis at the origin, \((0, 0)\).

Identifying these points helps in sketching the basic outline of the graph.

End Behavior

End behavior describes how a polynomial function behaves as the input values become very large or very small.For our function \(f(x) = x^3(x - 2)\), the degree of the polynomial is 4 (when expanded, the leading term is \(x^4\)). The degree of the polynomial plays a critical role in determining end behavior:

• Since the leading term \(x^4\) is positive and the degree is even, as \(x \to +\infty\), \(f(x) \to +\infty\). Similarly, as \(x \to -\infty\), \(f(x) \to +\infty\).

This indicates that as you move farther along the x-axis in either direction, the function's value rises indefinitely. Observing the end behavior is useful for assessing the function's vertical trends and assisting in graphing the function effectively.

Graphing Calculators

Graphing calculators are invaluable tools for students when exploring polynomial functions. They can quickly plot the graph of a function and provide visual insight into intercepts and end behavior. To graph the function \(f(x) = x^3(x - 2)\):

• Input the function into the calculator. Ensure you adjust your window settings appropriately to view the intercepts and the general shape.
• Identify the key features of the graph, such as where it crosses the axes and the shape in relation to the end behavior.

Using a graphing calculator facilitates a deeper understanding of the function's structure and helps you verify manual calculations and predictions.

Factoring Polynomials

Factoring is a central technique in handling polynomial functions. It simplifies the solution of equations and aids in graphing.The given function \(f(x) = x^3(x - 2)\) is already factored, showcasing its roots clearly. Here are some benefits of factoring:

• Simplifies finding intercepts: By expressing \(f(x)\) as a product of its factors, we easily find the x-intercepts, as setting each factor to zero provides the intercepts directly.
• Aids in graph sketching: Knowing intercepts and factored form allows us to sketch the graph more intuitively, understanding points of tangency and how the graph moves between intercepts.

By examining polynomials in their factored form, students can more efficiently analyze critical points and behavior of the polynomial, making complex algebraic expressions more approachable.