n the xy -plane, the graph of y = x^2

and the circle with
center (0,1) and radius 3 have how many points of intersection?

two solutions is correct, but the equation for a circle is:

(x-a)^2 + (y-b)^2 = r^2; where (a, b) is the center of the circle, so
(x-0)^2 + (y-1)^2 = 9

The points of intersection are:
(1.836377228, 3.372281323) and

(-1.836377228, -1.372281323)

To find the points of intersection between the graph of the equation y = x^2 and the circle with center (0,1) and radius 3, we need to solve both equations simultaneously.

Step 1: Set up the equations:
The equation of the circle can be written as:
(x - 0)^2 + (y - 1)^2 = 3^2
x^2 + (y - 1)^2 = 9

The equation of the graph is given as:
y = x^2

Step 2: Substitute the value of y in the equation of the circle:
x^2 + (x^2 - 1)^2 = 9

Step 3: Simplify the equation:
Expand the equation and combine like terms:
x^4 - 2x^2 + 1 + x^2 = 9
x^4 - x^2 - 8 = 0

Step 4: Solve the quadratic equation:
Let's solve the equation x^4 - x^2 - 8 = 0 for x. This equation can be factored as:
(x^2 - 4)(x^2 + 2) = 0

Setting each factor equal to zero gives us two sets of solutions:
x^2 - 4 = 0 and x^2 + 2 = 0

Solving for x in each equation gives us:
x = ± 2 and x = ± √2i

Step 5: Find the corresponding values of y:
Substituting the values of x into the equation y = x^2, we get:
For x = -2: y = (-2)^2 = 4
For x = 2: y = (2)^2 = 4

So, there are two points of intersection between the graph of y = x^2 and the circle with center (0,1) and radius 3. The points are (-2, 4) and (2, 4).

To find the number of points of intersection between the graph of y = x^2 and the circle with center (0,1) and radius 3, we need to determine the points where the two equations are equal.

First, let's express the equation of the circle in terms of x and y. The equation for a circle centered at (h, k) with radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

Substituting the values of the center (0,1) and radius 3, we have:

x^2 + (y - 1)^2 = 9

Now, we can set the equations of the circle and the parabola equal to each other:

x^2 + (y - 1)^2 = 9
y = x^2

Substituting the expression for y from the second equation into the first equation, we have:

x^2 + (x^2 - 1)^2 = 9

Simplifying and expanding the equation, we get:

x^4 - 2x^2 + 1 + x^4 - 2x^2 + 1 = 9
2x^4 - 4x^2 - 7 = 0

Now, we have a quartic equation in terms of x. We can solve this equation by factoring or using numerical methods to find the roots. Once we have the x-coordinates of the points of intersection, we can substitute them back into either the equation of the parabola or the equation of the circle to find the corresponding y-coordinates.

The number of points of intersection will be the number of real solutions to the equation 2x^4 - 4x^2 - 7 = 0. This equation may have zero, two, or four real solutions, so we need to solve it to determine how many points of intersection there are.

To find the solutions, you can use various numerical methods like graphing the equation, using a graphing calculator or software, or applying numerical root-finding methods such as Newton's method or the bisection method.

Once you have the x-values of the points of intersection, substitute them back into either the equation of the parabola or the equation of the circle to find the corresponding y-values. This will give you the complete set of points of intersection between the two curves.

circle: x^2 + y^2 = 9

parabola: y = x^2

using substitution
y + y^2 = 9
y^2 + y - 9 = 0

this quadratic will have 2 real solutions,
so there must be 2 intersection points

(make a quick sketch to verify this for yourself )