1.8 Common Graphs in Three Dimensions

Common Graphs in Three Dimensions

Introduction

Cylinder Surfaces

Quadric Surfaces

For each equation below, identify its graph in . Note that there are more graphs than equations, so some graphs will not be selected. To help you with identification, you might consider looking for $x$-intercepts, $y$-intercepts, and values of a variable for which the graph contains no points. -$\frac{x^2}{9}+\frac{y^2}{25}=1$-$\frac{x}{3}+\frac{y}{5}=1$-$\frac{y^2}{4}-\frac{x^2}{1}=1$-$x=2y^2$-$\frac{y^2}{9}-\frac{x^2}{25}=0$-$\frac{x^2}{1}-\frac{y^2}{4}=1$

Work through this exercise and explain your reasoning step by step.

$2x-y+1=0$ is called a linear cylinder surface.

$(x-1)^2+(y+2)^2=4$ is called a right-circular cylinder surface.

$\frac{x^2}{9}+\frac{z^2}{4}=1$ is called an elliptic cylinder surface.

$x^2-y^2=1$ is called a hyperbolic cylinder surface.

Find all $x$-, $y$-, and $z$-intercepts of $\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1$.

Find an equation for the curve given by the intersection of $\frac{x^2}{4} + \frac{y^2}{9} - \frac{z^2}{1} = 1$ with the $xy$-plane, the $yz$-plane, and the $xz$-plane. Draw a two-dimensional plot of each intersection.

For this activity, we will be looking at a variety properties that will help us draw a graph of the surface described by $\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1$.

Sketch each of these intersections on the proper fundamental planes in three dimensions.

Which of the following surface plots will correspond to $\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1$? You can determine this by comparing the features on your previous part to these options.

$\displaystyle\frac{x^2}{4}-\frac{y^2}{9}-\frac{z^2}{1}=1$

$\displaystyle\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{1}=1$

$\displaystyle\frac{x^2}{4}-\frac{y^2}{9}-\frac{z}{1}=0$