Chapter 11, Problem 54RE

a.

To determine

To draw: The graph of the following family of parabola for $y=k{x}^2,k=\frac{1}{2},1,2,4$

Explanation of Solution

Given information:

  $y=k{x}^2$ for values of $k=\frac{1}{2},1,2,4$

Graph:

  

  

  

  

Interpretation:

From the above all $4$ it is observed that domain of function is R and range is $[0,\infty ]$

b.

To determine

To Find: The foci of all the $4$ parabola is symmetrical about y − axis.

Answer to Problem 54RE

  $\begin{array}{l}\left(0,\frac{1}{8}\right)\text{ for }y=\frac{1}{2}{x}^2\\ \left(0,\frac{1}{4}\right)\text{ for }y=x^2\\ \left(0,\frac{1}{8}\right)\text{ for }y=2x^2\\ \left(0,\frac{1}{16}\right)\text{ for }y=4x^2\end{array}$

Explanation of Solution

Given information:

The equation of $4$ parabola's $y=\frac{1}{2}{x}^2,x^2,2x^2,4x^2$

Concept Used:

For any general parabola $y=a{x}^2$ the focus of this parabola lies on $\left(0,\frac{1}{4a}\right)$ point on the y axis.

Calculation:

For $y=\frac{1}{2}{x}^2\Rightarrow a=\frac{1}{2}$

Foci $\left(0,\frac{1}{4a}\right)=\left(0,\frac{1}{4\times \frac{1}{2}}\right)\Rightarrow \left(0,\frac{1}{2}\right)$

For $y=x^2\Rightarrow a=1$

Foci $\left(0,\frac{1}{4a}\right)=\left(0,\frac{1}{4}\right)$

For $y=2x^2$

Foci $\left(0,\frac{1}{4a}\right)\Rightarrow \left(0,\frac{1}{8}\right)$

For $y=4x^2$

Foci $\left(0,\frac{1}{4a}\right)\Rightarrow \left(0,\frac{1}{16}\right)$

Conclusion:

The required foci values symmetrical about y − axis of parabolas are obtained.

c.

To determine

To Find: How the location of focus changes as k increases.

Answer to Problem 54RE

As it can analyze from the graph being plotted we can see that as value of k increase the spread of parabola around y axis becomes closer to y axis.

Explanation of Solution

Given information:

The equation of curves parabola's $y=\frac{1}{2}{x}^2,x^2,2x^2,4x^2$

Graph:

  

  

  

  

Concept Used:

As it can analyze from the graph being plotted we can see that as value of k increase the spread of parabola around y axis becomes closer to y axis.