Problem 34 - A Graphical Approach to College Algebra

Question

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}-8 x^{2}+16 \\\&=(x+2)^{2}(x-2)^{2}\end{aligned}$$

Step-by-Step Solution

Answer

The graph of \( P(x) =(x+2)^2(x-2)^2 \) is a W-shaped curve, symmetric about the y-axis with intercepts at \( x = -2 \) and \( x = 2 \).

Step 1: Understand the Function

The function given is \( P(x) = x^4 - 8x^2 + 16 \). It can be factored into \( (x+2)^2(x-2)^2 \). Recognize this as a polynomial of degree 4. Since it’s in the form of a perfect square, it indicates roots with even multiplicity.

Step 2: Identify Key Features

The function \( P(x) = (x+2)^2(x-2)^2 \) has roots at \( x = -2 \) and \( x = 2 \), both with a multiplicity of 2, meaning the graph touches the x-axis at these points but does not cross it. The leading term \( x^4 \) indicates that the graph opens upwards and is symmetrically shaped because all the powers are even.

Step 3: Determine Behavior at Intercepts

At both intercepts \( x = -2 \) and \( x = 2 \), the graph only touches the axis and turns back upwards, because they are points of tangency (due to even multiplicity).

Step 4: Consider End Behavior

For large values of \( x \), the function \( P(x) \sim x^4 \) behaves like \( y = x^4 \). This means as \( x \to \, \pm \infty \), \( P(x) \to \, \infty \). The graph rises on both ends.

Step 5: Sketch the Graph

Draw the x-axis and y-axis. Plot the turning points at \( x = -2 \) and \( x = 2 \). The graph should touch these points and rise again on either side, forming a W-like shape. The symmetry about the y-axis can be noted since it is an even function.

Key Concepts

Polynomial Degree

The degree of a polynomial is an essential concept in graphing polynomial functions. The degree of the polynomial tells us the maximum number of roots it can have and, generally, the behavior of the graph at its extremities. For example, in the given function \( P(x) = x^4 - 8x^2 + 16 \), we notice that the highest power of \( x \) is 4. Therefore, the polynomial degree is 4. This is a critical piece of information as it helps predict the graph's shape.

For a degree 4 polynomial:

• There can be up to 4 roots; however, based on its factorized form \( (x+2)^2(x-2)^2 \), this polynomial has exactly 2 unique roots -- \( x = -2 \) and \( x = 2 \).
• We observe that since the degree is even, the ends of the graph will rise or fall together.
• The number of turning points on the graph is at most the degree minus 1, which means up to 3 turning points for this polynomial.

Understanding the degree aids in predicting the general shape and the complexity of the polynomial's graph.

End Behavior

The end behavior of a polynomial graph describes how it behaves as \( x \) approaches positive or negative infinity. This behavior is largely determined by the sign and degree of the leading term, which is the term with the highest power of \( x \). In our polynomial \( P(x) = x^4 - 8x^2 + 16 \), the leading term is \( x^4 \).

For polynomials with an even degree:

• If the leading coefficient (the coefficient of the highest degree term) is positive, both ends of the graph will rise to infinity. This is true for \( P(x) \), as the coefficient of \( x^4 \) is 1, which is positive.
• If the leading coefficient were negative, it would mean both ends of the graph would fall towards negative infinity.

This prediction can be visualized for \( P(x) \). As \( x \to \pm \infty \), the function \( P(x) \to \infty \). Thus, the graph rises on both the far left and far right sides, consistent with a W-shaped curve that opens upwards.

Multiplicity of Roots

The multiplicity of a root is an indicator of how many times a particular root appears in a polynomial. It significantly influences how the graph intersects with or touches the x-axis at these roots. For \( P(x) = (x+2)^2(x-2)^2 \), we have roots at \( x = -2 \) and \( x = 2 \), each with a multiplicity of 2.

When a root has:

• An odd multiplicity, the graph will cross the x-axis at that root.
• An even multiplicity, the graph will touch the x-axis and turn around without crossing it.

In this exercise, because both roots have even multiplicities of 2, the graph touches the x-axis at \( x = -2 \) and \( x = 2 \) and bounces back upwards, creating points of tangency. This behavior is crucial in understanding the movement of the graph and its overall W-like appearance. Recognizing multiplicity helps in accurately plotting the graph's course at its roots.