By completing the square, put the equation 2 – 64y tử + 42 – 16z = –68 Into the standard form a(z-zo)2 + b(y-yo)2 + c(z-zo)²=d. Then, use your answer to sketch a graph of this quadric surface on paper. 00+00+0(0)²=0

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Completing the Square and Graphing Quadrics**

By completing the square, put the equation 

\[ x^2 - 64y^2 + z^2 + 4x - 16z = -68 \]

into the standard form 

\[ a(x - x_0)^2 + b(y - y_0)^2 + c(z - z_0)^2 = d. \]

Then, use your answer to sketch a graph of this quadric surface on paper.

\[ (\square)^2 + (\square)^2 + (\square)^2 = \square \]

*Instructions:*

1. **Identify coefficients** for \(x^2\), \(y^2\), and \(z^2\) in the given equation.
2. **Complete the square** for the \(x\) and \(z\) terms to transform the equation into standard form.
3. **Simplify** the equation into the desired standard form by rearranging and factoring.
4. **Sketch** the quadric surface using the standard form equation to identify the center, orientation, and axes.
5. **Interpret the graph** based on the coefficients \(a\), \(b\), and \(c\), which implies the type of quadric surface (e.g., ellipsoid, hyperboloid, etc.).

This exercise involves algebraic manipulation to visualize geometric structures, enhancing your understanding of three-dimensional surfaces.
Transcribed Image Text:**Completing the Square and Graphing Quadrics** By completing the square, put the equation \[ x^2 - 64y^2 + z^2 + 4x - 16z = -68 \] into the standard form \[ a(x - x_0)^2 + b(y - y_0)^2 + c(z - z_0)^2 = d. \] Then, use your answer to sketch a graph of this quadric surface on paper. \[ (\square)^2 + (\square)^2 + (\square)^2 = \square \] *Instructions:* 1. **Identify coefficients** for \(x^2\), \(y^2\), and \(z^2\) in the given equation. 2. **Complete the square** for the \(x\) and \(z\) terms to transform the equation into standard form. 3. **Simplify** the equation into the desired standard form by rearranging and factoring. 4. **Sketch** the quadric surface using the standard form equation to identify the center, orientation, and axes. 5. **Interpret the graph** based on the coefficients \(a\), \(b\), and \(c\), which implies the type of quadric surface (e.g., ellipsoid, hyperboloid, etc.). This exercise involves algebraic manipulation to visualize geometric structures, enhancing your understanding of three-dimensional surfaces.
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