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Chapter 1: Problem 32

Use a graphing utility to graph the equations and to approximate the \(x\) -intercepts. In approximating the \(x\) -intercepts, use a "solve" key or a sufficiently magnified view to ensure that the values you give are correct in the first three decimal places. Remark: None of the \(x\) -intercepts for these four equations can be obtained using factoring techniques.) $$y=8 x^{3}-6 x-1$$

Short Answer

Expert verified
Approximate the \( x \)-intercepts using a graphing utility with three decimal places.

Step by step solution

01

Understand the Equation

We are dealing with the cubic equation \( y = 8x^3 - 6x - 1 \). Our goal is to find the \( x \)-intercepts, where \( y = 0 \). This means solving the equation for \( x \) when \( 8x^3 - 6x - 1 = 0 \).
02

Graph the Equation

Use a graphing utility or calculator to plot the graph of the equation \( y = 8x^3 - 6x - 1 \). Make sure to properly scale the axes to view the curve distinctly. This curve will help us visually identify the \( x \)-intercepts.
03

Locate the x-intercepts

On the graph, the \( x \)-intercepts are the points where the curve crosses the \( x \)-axis, meaning these are the values of \( x \) for which \( y = 0 \). "Zoom in" on these points if necessary to observe them more closely.
04

Approximate x-intercepts

Using the "solve" function of the graphing utility, find the exact values or a sufficiently accurate approximation of the \( x \)-intercepts, precise to three decimal places. Common utilities offer options to trace or calculate roots directly on the graph.

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Key Concepts

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

Understanding X-intercepts
X-intercepts are vital in understanding the roots of any function. They are the points where the graph crosses the x-axis, meaning the output y is zero. For the cubic equation \( y = 8x^3 - 6x - 1 \), the x-intercepts are the solutions for \( x \) when \( y \) equals zero. To clarify, you seek the x-values that make the equation \( 8x^3 - 6x - 1 = 0 \) true.

Finding x-intercepts graphically involves:
  • Plotting the curve on a graph.
  • Identifying where the curve crosses the x-axis.
  • These crossing points are your x-intercepts.
In this specific example, traditional algebraic factoring methods do not apply, so graphical methods or numerical approximations become necessary.
Using a Graphing Utility
Graphing utilities are powerful tools for visualizing mathematical equations, and they make finding x-intercepts much simpler. A graphing utility can be a physical calculator or software on a computer or phone, capable of plotting complex equations.

Here's how you use a graphing utility to help:
  • Input the equation into the graphing utility: \( y = 8x^3 - 6x - 1 \).
  • Adjust the graphing window to comfortably view the curve.
  • Ensure the scale is set to expose behavior around the x-axis accurately.
  • Use built-in tools such as "zoom" to refine your view on potential x-intercepts.
By clearly visualizing the curve, the graphing utility provides insight into where x-intercepts are likely to be located.
Approximating Roots
When finding exact x-intercepts is impossible by simple means, approximating them is the next best option. Using a graphing utility, one can utilize a function called "solve" or "zero" to locate these intercepts with precision.

Steps to approximate roots with a graphing utility:
  • Make sure the graph is accurately displaying the x-intercepts.
  • Use the solve function to find approximate values for the x-intercepts.
  • Check the precision. Aim for solutions that are accurate to three decimal places.
Approximating roots requires careful zooming and solving techniques, ensuring the result is both accurate and reliable for analysis.

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Most popular questions from this chapter

Find an equation of the line with the given slope and \(y\) -intercept. (a) slope \(-4 ; y\) -intercept 7 (b) slope \(2 ; y\) -intercept \(3 / 2\)

Specify the center and radius of each circle. Also, determine whether the given point lies on the circle. $$(x+8)^{2}+(y-5)^{2}=13 ;(-5,2)$$

You \(\%\) need to recall the following definitions and results from elementary geometry. In a triangle, a line segment drawn from a vertex to the midpoint of the opposite side is called a median. The three medians of a triangle are concurrent; that is, they intersect in a single point. This point of intersection is called the centroid of the triangle. A line segment drawn from a vertex perpendicular to the opposite side is an altitude. The three altitudes of a triangle are concurrent; the point where the altitudes intersect is the orthocenter of the triangle. This exercise illustrates the fact that the altitudes of a triangle are concurrent. Again, we'll be using \(\triangle A B C\) with vertices \(A(-4,0), B(2,0),\) and \(C(0,6) .\) Note that one of the altitudes of this triangle is just the portion of the \(y\) -axis extending from \(y=0\) to \(y=6 ;\) thus, you won't need to graph this altitude; it will already be in the picture. (a) Using paper and pencil, find the equations for the three altitudes. (Actually, you are finding equations for the lines that coincide with the altitude segments.) (b) Use a graphing utility to draw \(\triangle A B C\) along with the three altitude lines that you determined in part (a). Note that the altitudes appear to intersect in a single point. Use the graphing utility to estimate the coordinates of this point. (c) Using simultaneous equations (from intermediate algebra), find the exact coordinates of the orthocenter. Are your estimates in part (b) close to these values?

Determine the center and the radius for the circle. Also, find the \(y\) -coordinates of the points (if any) where the circle intersects the \(y\) -axis. $$9 x^{2}+54 x+9 y^{2}-6 y+64=0$$

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-1 / x^{3}$$
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