Question

Sketch the graph of the function. $f(x, y)=9-x^2-9 y^2$

   Sketch the graph of the function.
 $f(x, y)=9-x^2-9 y^2$

Calculus

James Stewart 7th Edition

Chapter 14, Problem 29

Instant Answer

Solved by Expert Abigail Martyr

08/11/2020

Step 1

The function \( f(x, y) = 9 - x^2 - 9y^2 \) is a quadratic function in two variables. It can be rewritten as \( f(x, y) = 9 - x^2 - 9y^2 \), which is in the form of an elliptic paraboloid. This is because it has the general form \( z = c - ax^2 - by^2 \), where Show more…

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Sketch the graph of the function. $f(x, y)=9-x^2-9 y^2$

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Key Concepts

Multivariable Function Graphing

This concept involves plotting functions of two variables by assigning every point (x, y) in the domain a corresponding height f(x, y). The process typically includes analyzing the domain, range, and using methods such as surface plots and contour lines to visualize how the function behaves in three-dimensional space.

Quadratic Surfaces

These surfaces are defined by quadratic equations in two or three variables. Understanding quadratic surfaces involves recognizing the general form, symmetry properties, and identifying key features like vertices and axes. They include various shapes such as paraboloids, ellipsoids, and hyperboloids.

Elliptical Paraboloid

An elliptical paraboloid is a type of quadratic surface where every cross section parallel to one coordinate plane (usually the horizontal plane) is an ellipse. This structure has a distinct vertex—the highest or lowest point on the surface—and it opens upward or downward depending on the sign of the quadratic terms.

Contour Plots/Level Curves

Contour plots or level curves are graphical representations that show curves on which the function f(x, y) takes on constant values. They provide important insights into the behavior of the function by revealing the gradient and curvature, and they aid in understanding the overall geometry of the surface.

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