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Chapter 4: Problem 39

For the following exercises, plot a graph of the function. $$z=f(x, y)=\sqrt{x^{2}+y^{2}}$$

Short Answer

Expert verified
The graph is a three-dimensional conical surface centered at the origin.

Step by step solution

01

Understand the Function

The function given is \( z = f(x, y) = \sqrt{x^2 + y^2} \). This function is the formula for the distance from the origin to a point \((x, y)\) in the xy-plane. It represents a three-dimensional surface known as a conical surface, also called a cone with a circular cross-section.
02

Identify the Shape of the Function

The function \( z = \sqrt{x^2 + y^2} \) describes a conical surface where the value of \( z \) depends on the distance of the point \((x, y)\) from the origin. As \( x \) and \( y \) increase in magnitude from zero, \( z \) also increases, forming a circular cone that opens upward from the origin.
03

Choose Graphing Range

To plot the graph of the function, choose appropriate values for \( x \) and \( y \). Common ranges include \( x \) from -5 to 5 and \( y \) from -5 to 5 to adequately show the shape of the cone and its symmetry around the origin.
04

Plot Corresponding Points

Calculate \( z \) for selected pairs of \( x \) and \( y \) within the chosen range. For example, calculate \( z \) for \((x, y)\) pairs like (0,0), (3,4), (-4,3), etc. This will give points \((x, y, z)\) that lie on the surface of the cone.
05

Draw the Graph

Using the calculated points, plot \( z = \sqrt{x^2 + y^2} \) on a three-dimensional graph. The graph will form a cone with its vertex at the origin (0,0,0) and a circular base as it extends outward. Lines of equal distance \( z \) from the origin will form circles parallel to the xy-plane.

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Key Concepts

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

Conical Surface
A conical surface is a 3D geometric shape that looks like a cone. The function given, \( z = \sqrt{x^2 + y^2} \), describes such a surface where the height \( z \) above any point \((x, y)\) is determined by its distance from the origin. This equates to a circle in the xy-plane that extends upwards to form a cone.
  • Vertex: The cone's tip is at the origin \((0, 0, 0)\).
  • Axis: The axis of the cone runs along the z-axis.
  • Circular Cross-section: Any cross-section parallel to the xy-plane will be a circle.
As \( x \) and \( y \) move further from zero, the circles become larger, forming the cone's base farther from the origin.
3D Graphing
Graphing in three-dimensional space involves plotting values for three variables, typically \( x, y, \) and \( z \). The function \( z = \sqrt{x^2 + y^2} \) provides a perfect opportunity to explore 3D graphing.
  • Coordinate System: Use a Cartesian coordinate system with axes for \( x, y, \) and \( z \).
  • Graphing Range: Choose a range, e.g., \( x \) and \( y \) from -5 to 5, which provides a complete view of the shape.
  • Plot Points: For each pair of \( (x, y) \), calculate \( z \), giving points \((x, y, z)\) on the surface.
This results in a visual representation of a cone on a 3D plot, highlighting the symmetry and structure of the mathematical expression.
Distance Formula
The distance formula is a key mathematical tool and its 2D version is useful in our function. The expression \( \sqrt{x^2 + y^2} \) is the distance from the origin to the point \((x, y)\) in the xy-plane.
  • Origin: It references the fixed starting point \((0, 0)\) in 2D.
  • Purpose: Calculates how far a point is from the origin, effectively mapping the radius of our conical cross sections.
  • 3D Extension: In 3D, the general distance formula becomes more complex, but here provides heights for our function.
Understanding this helps in visualizing how \( z \) corresponds to radial distance, forming the cone shape.
Coordinate Systems
Coordinate systems are essential for mapping mathematical functions into a space that we can visualize. In this context, our function uses a Cartesian coordinate system, which is perfect for describing surfaces like cones.
  • Axes: Consists of three perpendicular axes - \( x, y, \) and \( z \).
  • Representation: Places points in a space relative to these axes where each axis reflects a dimension.
  • 3D Function Placement: Here, each point has coordinates \( (x, y, z) \), allowing for precise positioning of the surface.
Choosing the correct coordinate system is vital for accuracy in graphing and understanding the geometric representation of the cone.

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Most popular questions from this chapter

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