Question

think about the graphs of the equations in the following system.
$x^{2}+y^{2}=2$
$y = 2x^{2}-3$
which of the following describes the system?
circle with its center at $(0,0)$ and a radius of $sqrt{2}$; parabola opening up with its vertex at $(0,-3)$
circle with its center at $(0,0)$ and a radius of $sqrt{2}$; parabola opening up with its vertex at $(0,3)$
circle with its center at $(0,0)$ and a radius of 2; parabola opening up with its vertex at $(0,-3)$
circle with its center at $(0,0)$ and a radius of 2; parabola opening up with its vertex at $(0,3)$

Explanation:

• For the equation of a circle \(x^{2}+y^{2}=r^{2}\), comparing with \(x^{2}+y^{2}=2\), we have \(r = \sqrt{2}\) and center \((0,0)\).
• For the parabola \(y = ax^{2}+bx + c\), in \(y=2x^{2}-3\), \(a = 2>0\) (opens up), and using the vertex formula \(x=-\frac{b}{2a}=0\), substituting \(x = 0\) into \(y=2x^{2}-3\) gives \(y=-3\), so vertex is \((0,-3)\).

Answer:

circle with its center at \((0, 0)\) and a radius of \(\sqrt{2}\); parabola opening up with its vertex at \((0, -3)\)