Question

a ) Describe the transformations to the graph of y = x² to obtain y = -2 ( x + 5 )²- 3 .

b ) Graph y = x² . Then apply the transformations in part ( a ) to graph y = -2 ( x + 5 ) ² - 3 .

c ) Determine the domain and range of this transformed function .

Answer 1

a) Consider the transformed function $y=-2(x+5)^2-3$ from the parent function $y=x^2$

i) Multiply $x^2$ by 2 to obtain $h(x)=2x^2$ in order to stretch the curve vertically as shown in the figure below:

ii) Multiply $h(x)=2x^2$ by $-1$ to obtain the function $g(x)=-2x^2$ in order to reflect the curve about $x$ -axis as shown in the figure below:

iii) Shift left by 5 units in order to transformed the function $l(x)=-2(x+5)^2$ as shown in the figure below:

iv) Shift vertically down by 3 units to obtain the final transformed function $y=-2(x+5)^2-3$ as shown in the figure below:

b) The sketch of the graph in part (a) is as shown in the figure below:

c) Since, the function $y=-2(x+5)^2-3$ is defined for all $x$ in $R^2$ ,

So, the domain of the function is $(-\infin,\infin)$

The graph of the function $y=-2(x+5)^2-3$ opens downward, so the maximum value is $-3$ .

Therefore, the range of the function is $(-\infin,-3]$ .