Chapter 11.4, Problem 41E

a.

To determine

Sketch: the graph of the family of parabolas

Explanation of Solution

Given information: The equation is $x^2=4p\left(y+p\right)$ where $p=-2,-\frac{3}{2},-1,-\frac{1}{2},1,\frac{1}{2},\frac{3}{2},2$

Graph: The graph of $x^2=4p\left(y+p\right)$ where $p=-2,-\frac{3}{2},-1,-\frac{1}{2},1,\frac{1}{2},\frac{3}{2},2$ shown below

  

Interpretation: from the above graph it is observed that this is conclude for any P focus is at origin

b.

To determine

To prove: that each parabola in his family has its focus at the origin

Explanation of Solution

Given information: The equation is $x^2=4p\left(y+p\right)$ where $p=-2,-\frac{3}{2},-1,-\frac{1}{2},1,\frac{1}{2},\frac{3}{2},2$

Proof:

To prove that each parabola has focus at origin

The above graph in part(a) shows the point which are lie at the origin

To put the values of P in provided equation then get points which are lie is at origin

So, all parabolas is at focus at origin

c.

To determine

To describe: that the effect on the graph of moving the vertex closer to the origin

Explanation of Solution

Given information: The equation is $x^2=4p\left(y+p\right)$ where $p=-2,-\frac{3}{2},-1,-\frac{1}{2},1,\frac{1}{2},\frac{3}{2},2$

Vertex moves from the positive y-axis to negative y-axis

Hence the vertex moves along the y-axis