Combining Transformations; Completing the Square

[trulyfalse]

Hello PF!

Homework Statement

The graph of the function y = 2x² + x +1 is stretched vertically about the x-axis by a factor of 2, stretched horizontally about the y-axis by a factor of 1/3 and translated 2 units right and 4 units down. Write the equation of the transformed function

Homework Equations

y= af(1/b(x-h))+k

The Attempt at a Solution

First I completed the square of y = 2x² + x +1:
2x² + x +1 = y
2(x²+1/2x+1/16-1/16)+1=y
2(x+1/4)²+14/16=y

Then, using mapping notation I calculated what the new x and y coordinates would be on the transformed function:
(x,y) → (1/3x+2,2y-4)
Therefore, the point (-2,7) → (4/3,10)

Using the aforementioned equation I transformed the function:
a=2
b=1/3
h=2
k=-4

y=2(2(3(x-7/4)²)-25/8)

However, when I input the new function into my calculator I received the values (-2,7) → (4/3,53/9) which is demonstrated to be incorrect. Is there a way to solve this problem without putting the function in the form y= af(1/b(x-h))+k? If possible, can anyone show me how to derive the correct equation for the transformed function after completing the square of the function? Thanks!

 

[HallsofIvy]

Hello PF!

Homework Statement

The graph of the function y = 2x² + x +1 is stretched vertically about the x-axis by a factor of 2, stretched horizontally about the y-axis by a factor of 1/3 and translated 2 units right and 4 units down. Write the equation of the transformed function

Homework Equations

y= af(1/b(x-h))+k

The Attempt at a Solution

First I completed the square of y = 2x² + x +1:
2x² + x +1 = y
2(x²+1/2x+1/16-1/16)+1=y
2(x+1/4)²+14/16=y

Almost. 1- 1/16= 15/16, not 14/16.

Thern, using mapping notation I calculated what the new x and y coordinates would be on the transformed function:
(x,y) → (1/3x+2,2y-4)
Therefore, the point (-2,7) → (4/3,10)

where did the "1/3" come from? If you are comparing y= 2(x+ 1/4)²+ 15/16 to y= x² then x changes to x+ 1/4 and y changes to 2y+ 15/16.

Using the aforementioned equation I transformed the function:
a=2
b=1/3
h=2
k=-4

y=2(2(3(x-7/4)²)-25/8)

However, when I input the new function into my calculator I received the values (-2,7) → (4/3,53/9) which is demonstrated to be incorrect. Is there a way to solve this problem without putting the function in the form y= af(1/b(x-h))+k? If possible, can anyone show me how to derive the correct equation for the transformed function after completing the square of the function? Thanks!