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Chapter 2: Problem 63

How Many Solutions? This exercise deals with the family of equations $$x^{3}-3 x=k$$ (a) Draw the graphs of $$y_{1}=x^{3}-3 x \quad \text { and } \quad y_{2}=k$$ in the same viewing rectangle, in the cases \(k=-4\) \(-2,0,2,\) and \(4 .\) How many solutions of the equation \(x^{3}-3 x=k\) are there in each case? Find the solutions correct to two decimals. (b) For what ranges of values of \(k\) does the equation have one solution? two solutions? three solutions?

Short Answer

Expert verified
There are three solutions for k = -4, -2, 0, 2, and one solution for k = 4. Three solutions exist for -3 < k < 3, two for k = -3 or 3, and one for k 鈮� -3 or k 鈮� 3.

Step by step solution

01

Graph the Function for k = -4

We begin by plotting the graph of \(y_1 = x^3 - 3x\) and \(y_2 = -4\). The function \(x^3 - 3x\) has inflection points at \(x = 0, \pm \sqrt{3}\).\(y_2 = -4\) is a horizontal line. Their intersections represent solutions. For this case, we observe three intersection points on the graph.
02

Graph the Function for k = -2

Next, we plot \(y_1 = x^3 - 3x\) and \(y_2 = -2\). Again, \(y_2 = -2\) is a horizontal line. Visually identify the intersections. For \(k = -2\), we find three intersection points, indicating three solutions.
03

Graph the Function for k = 0

Now, we consider \(y_2 = 0\). The intersection points for \(y_1 = x^3 - 3x\) and the line \(y_2 = 0\) happen where the graph crosses the x-axis. There are three such points: \(x = 0\) and two additional points symmetric about the origin, totaling three solutions.
04

Graph the Function for k = 2

For \(k = 2\), graph \(y_2 = 2\). \(x^3 - 3x\) crosses this horizontal line at three distinct points. Thus, there are three solutions for \(k = 2\).
05

Graph the Function for k = 4

Finally, we analyze \(y_2 = 4\). Observing the intersection points of the graph, we see only one intersection, indicating a single solution.
06

Determine the Range of k for Different Numbers of Solutions

The cubic function's shape results in three real roots if the horizontal line \(y_2 = k\) is below the inflection points and above the local minimum and maximum. For one solution, \(k\geq 3\) or \(k\leq -3\); for two solutions, exactly at \(k = 3\) or \(k = -3\); for three solutions, \(-3 < k < 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Functions
Graphing functions is an essential mathematical technique that helps us visualize how equations behave visually. When we graph a function, we plot points that satisfy the equation and connect them to reveal the overall shape. This procedure provides us with a clear picture of the relationship between variables.
  • For the function \(y_1 = x^3 - 3x\), the graph represents a cubic equation featuring the typical S-shape or wave-like form inherent to polynomials of degree three.
  • The inflection points, where the graph changes curvature, occur at \(x = 0\) and \(x = \pm \sqrt{3}\).
  • To observe how the values of \(k\) affect solutions, we overlay the line \(y_2 = k\), which is simply a horizontal line.
By graphing \(y_1\) and \(y_2\), we can visually interpret how many times they intersect, which directly relates to the number of solutions for the original equation \(x^3 - 3x = k\). Whether it meets once, twice, or three times gives us crucial insights into the equation's behavior.
Solutions of Equations
When we solve the equation \(x^3 - 3x = k\), we are searching for the x-values where the function \(f(x) = x^3 - 3x\) takes on the particular value \(k\). Each solution corresponds to an intersection point on the graph of \(y_1\) with the line \(y_2 = k\).
  • If there are three intersections, the equation has three solutions.
  • Two intersections indicate that the equation has two solutions.
  • Finally, a single intersection means there is only one solution.
The number of solutions can vary across different values of \(k\). In cases where \(-3 < k < 3\), the cubic equation yields three real root solutions. Conversely, if \(k\) is greater than or equal to 3, or less than or equal to -3, the equation results in a single real root. Specifically at values \(k = 3\) and \(k = -3\), there are precisely two solutions; the function becomes tangent to the horizontal line at these points.
Intersection Points
Intersection points are the specific x-values where two graphs meet, representing solutions to the equation when graphed. Consider our main function \(y_1 = x^3 - 3x\) and the horizontal line \(y_2 = k\). Each intersection point signifies a solution where the equation \(x^3 - 3x = k\) holds true.
  • To find these points accurately, we observe where the curves intersect visually or calculate precisely using algebraic methods.
  • Graphing tools or a graphing calculator can prove particularly helpful in pinpointing these intersection points, especially to meet accuracy requirements like rounding to two decimal places.
  • Understanding the nature of these intersections helps in unveiling the behavior of the cubic function and clarifying complexities in its solutions.
Grasping the concept of intersection points prepares students better to handle broader mathematical problems involving functions and equations. The visual interplay between \(y_1\) and \(y_2\) through graphing renders a crucial understanding of the practical implications of equation solutions.

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