Problem 58 - Precalculus with Calculus Previews

Question

Use a graphing utility to examine the graph of the given polynomial function on the indicated intervals. \(f(x)=(x-5)^{2}(x+5)^{2} ;[-10,10],[-100,100],[-1000,1000]\)

Step-by-Step Solution

Answer

The graph is symmetric about the y-axis and shows local minima at \(x = 5\) and \(x = -5\) on each interval.

Step 1: Understand the Polynomial Function

The given polynomial function is \(f(x) = (x-5)^2(x+5)^2\). It is a quartic polynomial, which means it has a degree of four and is generally expected to be a smooth curve that can have up to three turning points.

Step 2: Identify Critical Points

Critical points of the function can be found by setting the derivative \(f'(x)\) to zero or finding values where the function is not defined. However, with this polynomial, none of the factor parts are undefined, so the critical points occur at the zeros of the derivative, related to the zeros of the original polynomial: \(x=5\) and \(x=-5\). These zeros are repeated and act as the roots of the function.

Step 3: Examine the Graph on Interval \([-10, 10]\)

On the interval \([-10, 10]\), observing the graph using a graphing utility, we notice that the function has local maximum at its endpoints \(x = -10\) and \(x = 10\) and local minimums around the zeros \(x = 5\) and \(x = -5\). The graph is symmetric with the y-axis as the axis of symmetry.

Step 4: Examine the Graph on Interval \([-100, 100]\)

On \([-100, 100]\), the behavior around the roots \(x = -5\) and \(x = 5\) remains the same, locally, but as we extend the interval, the ends of the polynomial rise steeply upward due to the degree of the polynomial being even.

Step 5: Examine the Graph on Interval \([-1000, 1000]\)

With the interval extended to \([-1000, 1000]\), the graph continues to trend upwards rapidly with a similar symmetrical pattern around the center at \(x=0\). The boxy pattern around the roots \(x=5\) and \(x=-5\) becomes less discernible from a zoomed-out view, making the overall quartic pattern more pronounced.

Key Concepts

Quartic Polynomial

A quartic polynomial is a polynomial of degree four. It is typically represented as \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\), where \(a eq 0\). For our specific function, \(f(x) = (x-5)^2(x+5)^2\), we can expand it to reveal the full polynomial form, \(x^4 - 10x^2 + 25\).
Quartic polynomials are interesting because they can have up to three turning points. These polynomials are generally smooth, continuous curves that are symmetric in some way, depending on their leading coefficient.
Keep in mind that when analyzing a quartic polynomial, observing the pattern and behavior of the graph at different intervals can reveal much about these turning points and the overall shape of the curve.

Critical Points

Critical points of a function occur where the derivative is zero or undefined, indicating potential maxima, minima, or points of inflection. For polynomial functions like quartics, the critical points are found by setting the derivative, \(f'(x)\), to zero.
After finding the derivative of our example function, \(f(x) = (x-5)^2(x+5)^2\), we identify that the critical points are \(x = 5\) and \(x = -5\). These are where the function reaches local minima or maxima.
Because the original polynomial has repeated roots at these critical points, the graph tends to "flatten" or touch the x-axis at these points, rather than cross it. They are essential in shaping the graph and determining the symmetry.

Graphing Utility

A graphing utility is a valuable tool for visualizing polynomial functions and understanding their behavior over various intervals. For the quartic polynomial \(f(x) = (x-5)^2(x+5)^2\), using a graphing utility can help examine its shape, critical points, and symmetry.
By plotting the function on intervals like \([-10, 10]\), \([-100, 100]\), and \([-1000, 1000]\), we observe how the graph behaves locally versus globally. For instance, close to the roots \(x = 5\) and \(x = -5\), the graph exhibits local characteristics, whereas the behavior far from these points is dominated by the steep upward trend typical of quartic functions.
Utilizing a graphing utility allows us to adjust the view and gain insights into particular sections of the polynomial that would be challenging to conceptualize through analytical methods alone.

Symmetric Graph

The symmetry of a graph can greatly enhance our understanding of the function's properties. The example quartic polynomial \(f(x) = (x-5)^2(x+5)^2\) is symmetric about the y-axis. This is due to its even degree and the repeated roots, which make the function behave identically on both sides of the y-axis.
When analyzing a symmetric graph, it's crucial to locate the axis of symmetry, which in this case is \(x = 0\). This axis helps in predicting the behavior and the symmetry patterns of the graph, making it easier to conjecture about parts of the graph that are not visible in certain views.
Symmetrical shapes simplify understanding of critical elements like turning points, as the behavior is mirrored, providing a consistent pattern across the polynomial function.