# Graph sketching

• Jun 2nd 2007, 11:26 PM
margaritas
Graph sketching
The question is: Sketch 4x^2 + y^2 - 2y = 8 on a clearly labelled diagram, making clear the main relevant features of the graph.

Can I manipulate it such that it has an equation of an ellipse i.e. x^2/(3/2)^2 + (y-1)^2/3^2 = 1 ?

Or should I just make y the subject i.e. y = 1 +or- sq root (9-4x^2) then sketch using my GC?

Both give different sketches so which is the correct method or is there another way to do this question?
• Jun 3rd 2007, 04:21 AM
topsquark
Quote:

Originally Posted by margaritas
The question is: Sketch 4x^2 + y^2 - 2y = 8 on a clearly labelled diagram, making clear the main relevant features of the graph.

Can I manipulate it such that it has an equation of an ellipse i.e. x^2/(3/2)^2 + (y-1)^2/3^2 = 1 ?

Or should I just make y the subject i.e. y = 1 +or- sq root (9-4x^2) then sketch using my GC?

Both give different sketches so which is the correct method or is there another way to do this question?

Both methods will work (assuming you've solved the equations correctly; I didn't bother to check). The ellipse form tells you more than just plugging it in on the graphing calculator, though. (Such as the lengths of the axes, the foci, etc.)

-Dan
• Jun 3rd 2007, 05:09 AM
margaritas
Oh ok thanks! Could somebody help check my equations for each of the two methods? 'Cos I don't seem to get the same answer.
• Jun 3rd 2007, 05:30 AM
Soroban
Hello, margaritas!

Quote:

Sketch $4x^2 + y^2 - 2y \:=\:8$
Complete the square: . $4x^2 + y^2 - 2y + 1 \:=\:8 + 1$

Then we have: . $4x^2 + (y - 1)^2 \:=\:9$

Divide by 9: . $\frac{x^2}{\frac{9}{4}} + \frac{(y-1)^2}{9}\:=\:1\quad\Rightarrow\quad \frac{x^2}{\left(\frac{3}{2}\right)^2} + \frac{(y-1)^2}{3^2}\:=\:1
$

Solve for $y$

We have: . $4x^2 + (y-1)^2\:=\:9\quad\Rightarrow\quad(y-1)^2\:=\:0-4x^2$

. . $y - 1\:=\:\pm\sqrt{9 - 4x^2}\quad\Rightarrow\quad y \:=\:1\pm\sqrt{9-4x^2}$