Solve for the solution set of y >= x^2 and (x-1)^2 + (y+2)^2 <= 1 graphically.Show step by step process to explain the solution and answer

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labrat256 | (Level 2) Adjunct Educator

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To solve this graphically is fairly straight forward. First, completely ignore the inequalities and only plot the boundary. This is easy for the graph `y=x^2`

but in plotting the graph of `(x-1)^2+(y+2)^2=1`  you need to know that equations of the form `(ax-b)^2+(cy-d)^2=r`  are ellipses, or ovals. If a=c=1, then the equations form a circle of radius r. The centre of this circle is at point (b,d).

So, you have a circle with radius 1 at (1,-2) and parabola. Then plot them on the same graph.

Once you have plotted the boundaries you need to work out which areas of the graph satisfy the inequalities and which don't. The areas which satisfy the inequalities will always be bounded by one of the plotted graphs so to work out where satisfy the inequalities you will need to test some points. 

First, try inside the circle at point (1,-2). This satisfies the second inequality as

`(x-1)^2+(y+2)^2<=1`

`(1-1)^2+(y-2)^2=0+0<=1`

but it doesn't satisfy the first inequality as

`y>=x^2`

but

`-2<=1^2`

so we know that to satify the first inequality, it needs to be within the parabola and to satisfy the second, it needs to be in the circle. Now you shade them in on your graph and that shows graphically where the inequalities hold!

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