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 .

Accepted Answer

y = x² to obtain y = -2 ( x + 5 )²- 3

_y _= (_x_ − 6)(_x_ − 4)

When_ x _= 0, _y _= 24,

so the _y_-intercept is 24.

When _y _= 0,_ x _= 4 or_ x _= 6,

so the_ x_-intercepts are 4 and 6.

Taking the average of the_ x_-intercepts,

_x _= 5 is the axis of symmetry.

When_ x _= 5, _y _= (5 − 6)(5 − 4) = −1,

so the vertex_ _is (5, −1).

b

_y _= −_ x_2 +_ x _+ 6

This is an upside-down parabola.

When_ x _= 0, _y _= 6, so this is the _y_-intercept.

_y_= −_ x_2 +_ x _+ 6= −(_x_2 −_ x _− 6)= −(_x _−
3)(_x _+ 2)

So the_ x_-intercepts are 3 and −2.

Taking their average, the axis of symmetr_y _is_ x _= .

When_ x _= , _y _= − +  + 6 = 6 so the vertex_ _is , 6.

c

_y _= 5_x_2 − 20_x _+ 15

When_ x _= 0 then _y _= 15.

_y_= 5_ x_2 − 20_ x _+ 15= 5 (_x_2 − 4_x _+ 3 )= 5 (_x _−
3)(_ x _− 1)

So the two_ x_-intercepts are_ x _= 1 and_ x _= 3.

Hence, the axis of symmetr_y _is_ x _= 2 and the vertex_ _is (2,
−5).