Chapter 9.3, Problem 39HP

To determine

Describe how the values of a and c affect the graphical and tabular representation for the functions $y=a{x}^2;y=x^2+c;y=a{x}^2+c$

Explanation of Solution

Given:

Describe how the values of a and c affect the graphical and tabular representation for the functions $y=a{x}^2;y=x^2+c;y=a{x}^2+c$

Concept Used:

  $\begin{array}{l}\text{Vertical}\text{Stretch}\text{and}\text{Vertical}\text{Compression:}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{a}\text{>1}\text{,}\text{Srtetch}\text{the}\text{graph}\text{of}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\text{.}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{0}\text{<}\text{a}\text{<1}\text{,}\text{Compress}\text{the}\text{graph}\text{of}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\end{array}$

For $y=a{x}^2$ , the value of a stretches the parent graph vertically by a factor of a.

If $0<a<1$ , vertically compress the parent graph by a factor of a .

When $a<0$ or negative, the graph is reflected through the x − axis.

For $y=x^2+c$ , the parent graph is translated up if c is positive and translated down if c is negative. In Tabular form the value of c is always added to y.

For $y=a{x}^2+c$ , the graph will either be stretched or compressed according to the value of a and then will be translated up or down according to the value of c. In Tabular form the value of c is always added to y.

Calculation:

Example: the function $y=x^2$

x	$y=x^2$	(x, y)
− 4	16	(− 4,16)
− 2	4	(− 2, 4)
0	0	(0, 0)
2	4	(2, 4)
4	16	( 4 , 16)

  

Example: the function: $y=x^2+2$

x	$y=x^2+2$	(x, y)
− 4	18	(− 4,18)
− 2	6	(− 2, 6)
0	2	(0, 2)
2	6	(2, 6)
4	18	( 4 , 18)

  

Example: the function: $y=2x^2+2$

x	$y=2x^2+2$	(x, y)
− 4	34	(− 4,34)
− 2	10	(− 2, 10)
0	2	(0, 2)
2	10	(2, 10
4	34	( 4 , 34)

  

y=x2, $y=x^2+2$, $y=2x^2+2$ all are in same graph