Systems of Equations and Graphical Interpretations

Hi kenan, let's solve this problem involving a system of equations with a circle and a parabola. We will start by finding the original equations from the graph.

First, let's look at the circle. Its center is at the origin zero zero, and it passes through three on the x axis and three on the y axis. This means the radius r is three.

Since r equals three, r squared is nine. So, the equation of the circle is x squared plus y squared equals nine.

Now for the parabola. Notice the vertex is at zero negative three. Using the vertex form, y equals a times x squared minus three.

The parabola passes through the point one negative five. Let's plug this in to find a. Negative five equals a times one squared minus three.

Solving for a, we add three to both sides to get a equals negative two. So the parabola's equation is y equals negative two x squared minus three.

In part b, we need to shift this parabola vertically to get different numbers of intersection points with the circle. Our original parabola and circle touch at exactly one point, the vertex zero negative three.