Problem 58 - 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. $$f(x)=(x-20)^{2}(x+30)^{2}$$

Step-by-Step Solution

Answer

The function \(f(x) = (x-20)^{2}(x+30)^{2}\) has roots at \(x=20\) and \(x=-30\) with multiplicity 2, and both ends of the graph point upwards.

Step 1 - Identify the polynomial and its degree

The given polynomial function is \(f(x) = (x-20)^{2}(x+30)^{2}\). To determine the degree of the polynomial, expand the factors: \((x-20)^2 = x^2 - 40x + 400\) and \((x+30)^2 = x^2 + 60x + 900\). When multiplying the expanded forms: \(f(x) = (x^2 - 40x + 400)(x^2 + 60x + 900)\). This results in a polynomial of degree 4 (since the highest power of x after expanding will be \(x^4\)).

Step 2 - Determine the roots and their multiplicities

The roots of the polynomial can be found by setting each factor equal to zero: \(x-20=0\) gives \(x=20\) and \(x+30=0\) gives \(x=-30\). Each root has a multiplicity of 2 because each factor is squared. Hence, the roots are \(x=20\) and \(x=-30\), both with multiplicity 2.

Step 3 - Determine the end behavior of the polynomial

The leading term of a polynomial dictates the end behavior. Since this is a degree 4 polynomial with a positive leading coefficient (after expanding), the ends of the graph will both point upwards. Hence, as \(x\) approaches \(\text{infinity}\), \(f(x)\) approaches \(\text{infinity}\) and as \(x\) approaches \(-\text{infinity}\), \(f(x)\) also approaches \(\text{infinity}\).

Step 4 - Graph using a calculator and sketch the graph

Graph the polynomial function \(f(x) = (x-20)^{2}(x+30)^{2}\) using a graphing calculator. Observe the key features: - The graph touches the x-axis at \(x=20\) and \(x=-30\) and rebounds upward due to the even multiplicity. - The end points of the graph go to \(\text{infinity}\). Use these observations to sketch the graph, marking the key roots and noting the end behavior.

Key Concepts

Polynomial Degree

Understanding the degree of a polynomial is crucial for graphing it accurately. The degree of a polynomial is the highest power of the variable in the function. For example, in the polynomial function \[f(x) = (x-20)^2 (x+30)^2\], we can determine the degree by expanding the factors:

• \((x-20)^2 = x^2 - 40x + 400\)
• \((x+30)^2 = x^2 + 60x + 900\)

When you multiply these expanded forms, the highest power of x is \(x^4\), thus making it a degree 4 polynomial. The degree not only tells us about the highest power of x but also gives us information about the shape and the number of turning points the graph may have.

Roots and Multiplicities

Roots (or zeroes) are the values of x where the polynomial equals zero. Multiplicities refer to the number of times a particular root is repeated. For the given polynomial \[f(x) = (x-20)^2 (x+30)^2\], we find the roots by setting each factor to zero:

• \(x-20=0\) gives the root \(x=20\)
• \(x+30=0\) gives the root \(x=-30\)

Both of these factors are squared, meaning they have a multiplicity of 2. This has a visual impact on the graph. When the graph touches these roots, it will 'bounce off' rather than cross the x-axis due to the even multiplicity.

End Behavior of Polynomials

The end behavior of a polynomial describes how the function behaves as x approaches positive or negative infinity. For the polynomial \[f(x) = (x-20)^2 (x+30)^2\], which is a degree 4 polynomial with a positive leading coefficient, we observe the following behavior:

• As \(x\) approaches \(+\text{infinity}\), \(f(x)\) also approaches \(+\text{infinity}\).
• As \(x\) approaches \(-\text{infinity}\), \(f(x)\) once again approaches \(+\text{infinity}\).

This means that both ends of the graph will point upwards. The behavior is dictated by the polynomial's degree and the sign of the leading coefficient. Understanding this helps in sketching the correct shape of the polynomial graph.

Graphing Polynomial Functions

Plotting polynomial functions involves several steps to ensure accuracy:

• Identify the degree and leading coefficients.
• Determine the roots and their multiplicities.
• Examine the end behavior.

For our polynomial \[f(x) = (x-20)^2 (x+30)^2\], use the steps above to create a sketch. Mark the roots \(x=20\) and \(x=-30\). Note the behavior at these roots, where the graph touches and rebounds from the x-axis due to their even multiplicities. Then, consider the end behavior, ensuring both sides of the graph point upwards. These steps will allow you to create a graph that represents the polynomial accurately.

Using Graphing Calculators

Graphing calculators are valuable tools for visualizing polynomial functions. For example, input \[f(x) = (x-20)^2 (x+30)^2\] into your graphing calculator. Observe the displayed graph to identify key features:

• The roots, where the graph touches the x-axis.
• The points where the graph rebounds due to even multiplicities.
• The end behavior where both ends of the graph point upwards.

Using the graphing calculator as a guide, sketch the polynomial function on paper. This allows you to verify your manual calculations and provides a visual aid for understanding the polynomial's behavior.