Preview Activity 5.2.1.
Point your browser to the Desmos worksheet at http://gvsu.edu/s/0zy. There you’ll find a degree \(4\) polynomial of the form \(p(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\text{,}\) where \(a_0, \ldots, a_4\) are set up as sliders. In the questions that follow, you’ll experiment with different values of \(a_0, \ldots, a_4\) to investigate different possible behaviors in a degree \(4\) polynomial. Note that we require \(a_4 \ne 0\) in order to ensure \(p\) is a degree \(4\) polynomial.
(a)
Recall from the definition of a polynoimal function what we mean by a zero of the polynomial. Give examples of values for \(a_0, \ldots, a_4\) that lead to that largest number of zeros for \(p(x)\text{.}\)
(b)
What other numbers of zeros are possible for \(p(x)\text{?}\) Said differently, can you get each possible number of fewer zeros than the largest number that you found in (a)? Why or why not?
(c)
We say that a function has a turning point if the function changes from decreasing to increasing or increasing to decreasing at the point. For example, any quadratic function has a turning point at its vertex.
What is the largest number of turning points that \(p(x)\) (the function in the Desmos worksheet) can have? Experiment with the sliders, and give examples of values for \(a_0, \ldots, a_4\) that lead to that largest number of turning points for \(p(x)\text{.}\)
(d)
What other numbers of turning points are possible for \(p(x)\text{?}\) Can it have no turning points? Just one? Exactly two? Experiment and explain.
(e)
What long-range behavior is possible for \(p(x)\text{?}\) Said differently, what are the possible results for \(\displaystyle \lim_{x \to -\infty} p(x)\) and \(\displaystyle \lim_{x \to \infty} p(x)\text{?}\)
(f)
What happens when we plot \(y = a_4 x^4\) in Desmos and compare \(p(x)\) and \(a_4 x^4\text{?}\) How do they look when we zoom out? (Experiment with different values of each of the sliders, too.)
