Problem 49 - Calculus A New Horizon

Question

In each part: (i) Make a conjecture about the behavior of the graph in the vicinity of its \(x\) -intercepts. (ii) Make a rough sketch of the graph based on your conjecture and the limits of the polynomials as \(x \rightarrow+\infty\) and \(x \rightarrow-\infty .\) (iii) Compare your sketch to the graph generated with a graphing utility. (a) \(y=x(x-1)(x+1) \quad\) (b) \(y=x^{2}(x-1)^{2}(x+1)^{2}\) (d) \(y=x(x-1)^{5}(x+1)^{4}\) (c) \(y=x^{2}(x-1)^{2}(x+1)^{3}\)

Step-by-Step Solution

Answer

(a) Crosses at all intercepts; (b) touches all intercepts; (d) crosses at 0 only; (c) touches except at -1.

Step 1: Find the x-intercepts for each expression

To find the x-intercepts, set each expression equal to zero and solve for \(x\). For (a) \(y=x(x-1)(x+1)\):- \(x=0\), \(x=1\), \(x=-1\). For (b) \(y=x^{2}(x-1)^{2}(x+1)^{2}\):- \(x=0\), \(x=1\), \(x=-1\). For (d) \(y=x(x-1)^{5}(x+1)^{4}\):- \(x=0\), \(x=1\), \(x=-1\). For (c) \(y=x^{2}(x-1)^{2}(x+1)^{3}\):- \(x=0\), \(x=1\), \(x=-1\).

Step 2: Make a conjecture about behavior around x-intercepts

For each part, observe the multiplicity of each \(x\) value:- A zero of odd multiplicity indicates the graph crosses the x-axis.- A zero of even multiplicity indicates the graph touches but does not cross the x-axis.(a) \(x=0, 1, -1\) all have multiplicity 1, so the graph crosses the x-axis at each point.(b) \(x=0, 1, -1\) all have multiplicity 2, so the graph touches the x-axis but does not cross at each point.(d) \(x=0\) has multiplicity 1, so it crosses. \(x=1\) has multiplicity 5, it will touch but return. \(x=-1\) has multiplicity 4, it will only touch.(c) \(x=0, 1\) have multiplicity 2, the graph will only touch. \(x=-1\) has multiplicity 3, so it will cross.

Step 3: Sketch the graph based on conjecture and end behavior

Determine the end behavior of each polynomial by examining the leading term.(a) Given the leading term \(x^3\), as \(x \to +\infty\) or \(-\infty\), the graph will go to \(+\infty\) and \(-\infty\) respectively.(b) Given \(x^6\), as \(x \to \pm\infty\), the graph rises to \(+\infty\).(d) With leading term of \(x^{10}\), as \(x \to \pm\infty\), the graph rises to \(+\infty\).(c) Leading term is \(x^7\), as \(x \to \pm\infty\), the graph will go to \(+\infty\) and \(-\infty\) respectively.Now sketch by plotting x-intercepts and drawing the general behavior.

Step 4: Compare with graphing utility

Use a graphing tool to plot each equation and compare the sketch.a. Sketch crossings at all intercepts should match graph.b. Only touches at the intercepts and all end behaviors matching should.d. Graph should cross \(x=0\), plummets and rebounds around \(x=1\), and stays positive.c. \(x=0, 1\) touches only, \(x=-1\) should cross.

Key Concepts

Multiplicity

Multiplicity refers to the number of times a particular root occurs in a polynomial. It tells us how the graph behaves at the x-intercepts, where the polynomial function equals zero. To find the multiplicity of a root, look at the exponent of the factor associated with that root.

• If the multiplicity is odd, the graph of the polynomial will cross the x-axis at the intercept.
• If the multiplicity is even, the graph will touch the x-axis but not cross it.

For example, consider the polynomial function \(y = x(x-1)(x+1)\). Each factor \((x), (x-1),\) and \((x+1)\) appears once, so each has a multiplicity of 1. This means the graph will cross the x-axis at \(x = 0, 1,\) and \(-1\). Understanding multiplicity helps us predict and describe the polynomial's behavior at its roots.

X-intercepts

The x-intercepts of a function are the points where the graph crosses or touches the x-axis. These occur where the value of the function equals zero. To find the x-intercepts of a polynomial, you set the equation equal to zero and solve for x.

• For example, in the polynomial \(y = x(x-1)(x+1)\), to find the x-intercepts, set \(y = 0\).
• This gives the solutions \(x = 0\), \(x = 1\), and \(x = -1\).

At these intercepts, the graph will either touch or cross the x-axis based on the multiplicity of each factor. Recognizing these intercepts is a vital step in sketching the graph and understanding its behavior.

End Behavior

End behavior describes how the graph of a polynomial function behaves as x approaches positive or negative infinity. This is determined by the leading term of the polynomial, which has the highest power of x. By analyzing the degree and leading coefficient, you can predict the direction in which the graph heads at the ends of the x-axis.

• If the leading term has an odd exponent, like \(x^3\) in \(y = x(x-1)(x+1)\), then as \(x \to +\infty\), the graph goes to \(+\infty\) and as \(x \to -\infty\), the graph goes to \(-\infty\).
• If the leading term has an even exponent, such as \(x^6\) in \(y = x^2(x-1)^2(x+1)^2\), the graph will rise to \(+\infty\) on both sides.

Understanding end behavior helps in making rough sketches and conjectures about the graph outside of the x-intercepts.

Graphing Utility

A graphing utility is a tool that helps visualize the behavior of functions through graphical representations. It allows you to plot functions and compare your conjectures and sketches with accurate graphs.

• Graphing utilities can plot complex polynomial functions quickly and display attributes like x-intercepts and end behavior clearly.
• By using a graphing utility, you can verify if the behavior predicted by analyzing x-intercepts, multiplicity, and end behavior matches the actual graph.

For instance, after sketching the polynomial \(y = x(x-1)(x+1)\) based on predictions, use a graphing utility to check if the graph crosses at each intercept and if the end behavior aligns with your expectations. These tools are invaluable for learning and confirming the accuracy of your work.