Problem 57 - Algebra for College Students

Question

Sketch the graph of each polynomial function. First graph the function on a calculator and use the calculator graph as a guide. $$f(x)=(x-20)^{2}(x+30)$$

Step-by-Step Solution

Answer

The graph of \(f(x)=(x-20)^2(x+30)\) crosses the x-axis at \(x=-30\), touches and turns at \(x=20\), rises steeply, and follows the end behavior of \(x^3\).

Step 1 - Identify the type of polynomial and its degree

Identify that the given polynomial is of the form \(f(x)=(x-20)^2(x+30)\). Notice that it's a cubic polynomial (degree 3), as it can be expanded to get the leading term \(x^3\).

Step 2 - Find the roots (zeros) of the polynomial

Set \(f(x) = 0\) to find the roots. Solve \((x-20)^2(x+30) = 0\). The roots are \(x = 20\) (with multiplicity 2) and \(x = -30\).

Step 3 - Determine the end behavior

As \(x\) approaches \(+\text{infinity}\) or \(-\text{infinity}\), the behavior of a cubic polynomial \((x-20)^2(x+30)\) follows that of \(x^3\) since the highest degree term dominates. Thus, as \(x \to +\text{infinity}\), \(f(x) \to +\text{infinity}\), and as \(x \to -\text{infinity}\), \(f(x) \to -\text{infinity}\).

Step 4 - Analyze the multiplicity of the roots

The root \(x = 20\) has multiplicity 2, meaning the graph will touch the x-axis and turn around at \(x = 20\). The root \(x = -30\) has multiplicity 1, meaning the graph will cross the x-axis at \(x = -30\).

Step 5 - Calculate a few points

To better outline the graph, calculate the function value at a few points. For example, \(f(0)=(0-20)^2(0+30)=12000\) shows that the graph will rise steeply.

Step 6 - Sketch the graph

Start by marking the roots and plotting the points calculated. Sketch the overall shape based on the behavior at the roots, ensuring the end behavior and turning points match the given details. Use a calculator graph for fine adjustments.

Key Concepts

Polynomial Roots

Polynomial roots, also called 'zeros,' are the values of x that make the polynomial equation equal to zero. For the polynomial function given, \[f(x) = (x-20)^2(x+30)\]setting \[f(x) = 0\] gives us the roots:

• \(x = 20\)
• \(x = -30\)

Knowing these roots is critical because they help in sketching the graph accurately. Roots mark the points where the graph intersects the x-axis. In this example, the roots at x = 20 and x = -30 indicate where the polynomial function will touch or cross the x-axis.

Multiplicity of Roots

The term 'multiplicity' refers to the number of times a particular root appears in the polynomial. It affects the graph's behavior at those points. For instance, in the polynomial function, the root \(x = 20\) has a multiplicity of 2 and \(x = -30\) has a multiplicity of 1.

• If a root has an 'odd' multiplicity (e.g., 1), the graph 'crosses' the x-axis at that point. In our case, at \(x = -30\), the graph will cross the x-axis.
• If a root has an 'even' multiplicity (e.g., 2), the graph 'touches' the x-axis and turns around. For \(x = 20\), the graph will touch the x-axis and bounce back.

Understanding multiplicity is important for creating an accurate sketch, as it directly influences the shape of the graph at the roots.

End Behavior of Polynomials

End behavior describes how the polynomial function behaves as \(x\) approaches \(+\infty\) (positive infinity) and \(-\infty\) (negative infinity). For cubic polynomials like our function \[f(x) = (x-20)^2(x+30)\]the highest degree term (\(x^3\)) determines the end behavior. For our polynomial:

• As \(x \to +\infty\), \(f(x) \to +\infty\). This means the graph will rise to infinity on the right side.
• As \(x \to -\infty\), \(f(x) \to -\infty\). The graph will fall to negative infinity on the left side.

Recognizing the end behavior helps us sketch the graph's general direction as it moves far left or right.