Answer to Question #149193 in General Chemistry for aylol

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

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 = − x2 + x + 6

This is an upside-down parabola.

When x = 0, y = 6, so this is the y-intercept.

y= − x2 + x + 6= −(x2 − x − 6)= −(x − 3)(x + 2)

So the x-intercepts are 3 and −2.

Taking their average, the axis of symmetry is x = .

When x = , y = − +  + 6 = 6 so the vertex is , 6.

c 

y = 5x2 − 20x + 15

When x = 0 then y = 15.

y= 5 x2 − 20 x + 15= 5 (x2 − 4x + 3 )= 5 (x − 3)( x − 1)

So the two x-intercepts are x = 1 and x = 3.

Hence, the axis of symmetry is x = 2 and the vertex is (2, −5).

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 = − x2 + x + 6

This is an upside-down parabola.

When x = 0, y = 6, so this is the y-intercept.

y= − x2 + x + 6= −(x2 − x − 6)= −(x − 3)(x + 2)

So the x-intercepts are 3 and −2.

Taking their average, the axis of symmetry is x = .

When x = , y = − +  + 6 = 6 so the vertex is , 6.

c 

y = 5x2 − 20x + 15

When x = 0 then y = 15.

y= 5 x2 − 20 x + 15= 5 (x2 − 4x + 3 )= 5 (x − 3)( x − 1)

So the two x-intercepts are x = 1 and x = 3.

Hence, the axis of symmetry is x = 2 and the vertex is (2, −5).