y^2 = (x+3)^2: But how do i go about making sense of y^2 = (x+3)^2 ?

[apple2357]

I understand that y= x+3 draws a simple straight line.
But how do i go about making sense of y^2 = (x+3)^2 . Plotting on software, it draws me two lines presumably y= (x+3) and y= -(x+3).
But if i expand i get an implicit function y^2-x^2-6x-9 =0 and the idea of a straight line gets lost?
What is this beast?

 

[HallsofIvy]

I understand that y= x+3 draws a simple straight line.
But how do i go about making sense of y^2 = (x+3)^2 . Plotting on software, it draws me two lines presumably y= (x+3) and y= -(x+3).
But if i expand i get an implicit function y^2-x^2-6x-9 =0 and the idea of a straight line gets lost?
What is this beast?

I have no idea why you think that "the idea of a straight line gets lost".
Since the (x, y) values that satisfy \(\displaystyle y^2= (x+ 3)^2= x^2+ 6x+ 9\) also satisfy either y= x+3 or y= -(x+ 3) the graph is the two intersecting straight lines.

 

[mmm4444bot]

… an implicit function y^2 - x^2 - 6x - 9 = 0 and the idea of a straight line gets lost?

This equation can be read to imply a functional relationship between x and y. You are discussing one situation: y is a function of x.

Furthermore, you have already shown that there are two such linear functions. The lines are not "lost", when writing the equation above. They are "implied". :cool:

 

[sinx]

I understand that y= x+3 draws a simple straight line.
But how do i go about making sense of y^2 = (x+3)^2 . Plotting on software, it draws me two lines presumably y= (x+3) and y= -(x+3).
But if i expand i get an implicit function y^2-x^2-6x-9 =0 and the idea of a straight line gets lost?
What is this beast?

you will find that y²-x²-6x-9 =0 also yields a straight line, (actually two straight lines).
y=4x could be written as y²=16x² ; where y=+-4x; but y=-4x makes no sense. It is a line with a slope that is opposite from y=4x. So you have to make sense of the eqn after you square both sides.

 

[Denis]

y^2-x^2-6x-9 =0

x^2 + 6x + 9 - y^2 = 0
using quadratic:
x = -3 ± y

 

[apple2357]

Thanks for your thoughts all!