Translation/Reflection/Expansion/Compression of a graph

[mantha1974]

The instructions state:

Describe the graph of y = sqrt( 7 - x^2 - 6x ) relative to the graph of y = sqrt( x ). HINT: sqrt( 7 - x^2 -6x ) = sqrt( 7 - ( x^2 + 6x )); complete the square

This is my work thus far:

7 - ( X^2 + 6x ) = 0
- ( x^2 + 6x ) = -7
above mult by -1 (on both sides)
x^2 + 6x = 7
x^2 + 6x + 9 = 7 + 9
( x + 3 )^2 = 16
sqrt( ( x + 3 )^2) = sqrt( 16 )
x + 3 = +4
x = 1 and x = 7

so, now that i've completed the square.. I have no idea what to do next. My professor has succeeded in really confusing me. So frustrated. I entered the base graph and the new graph into my calculator and I can't even begin to describe what has been altered aside from the fact that it has been shifted horizontally to the left by 7. It appears to be a semi-circle now rather than a square root.

 

[wjm11]

y = sqrt( 7 - x^2 - 6x )

y^2 = 7 - x^2 - 6x
(x^2 + 6x + 9) +y^2 = 7 + 9
(x + 3)^2 + y^2 = 4^2

Perhaps now you recognize the equation for a circle of radius 4, centered at (-3,0). The original equation, y = sqrt( 7 - x^2 - 6x ), describes the top half of this circle.

The other equation in question, y = sqrt( x ), describes the top half of a sideways parabola.

Perhaps you are simply supposed to recognize the “top half” forms of various functions(?).

As an aside, you said, “x = 1 and x = 7.” This is incorrect; you have a sign error. Always check your answers by plugging them back into the original equation to see if they work.