Chapter 10.7, Problem 64AYU

To determine

To show:

The graph of the epicycloid is of the given form

Explanation of Solution

Given:

The equation of epicycloid

  $A{x}^2+C{y}^2+Dx+Ey+F=0$

Concept used:

The stabdard form of epicycloid equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $k=a{h}^2+bh+c\mathrm{.......................}\left(2\right)$

Where $\left(h,k\right)$ is the vertex

Calculation:

The stabdard form of epicycloid equation

  $y=a{x}^2+bx+c$

Putting $x=h,y=k$ in equation (1)

  $\begin{array}{l}k=a{h}^2+bh+c\\ {\left(y-k\right)}^2=4a\left(x-h\right)\mathrm{.......................}\left(2\right)\end{array}$

Where $\left(h,k\right)$ is the vertex

Draw the table

  $A{x}^2+C{y}^2+Dx+Ey+F=0$

Test one point in each of the region formed by the graph

If the point satisfies the function then shade the entire region to denote that every point in the region satisfies the function

$x-axis$	$-0.5$	$0$	$0.65$	$-1$
$y-axis$	$-0.75$	$-1$	$-2$	$-1$