Graph the system x^2 + y^2 <=49 and y <= 3 - x^2

Question

the class is pre-calculus.

Answer 1

The system x ² + y ² ≤ 49

y  ≤ 3 - x ²

To solve the system of equations,first graph the system.

The fact that these expressions are inequalities and not equations doesn’t change the general shape of the graph at all.

Therefore, you can graph these inequalities just as you would graph them if they were equations.

x ² + y ² = 49

(x - 0) ² + (y - 0) ² = 7 ²

Compare it to circle equation (x - h ) ² + (y - k ) ² = r  ²

The top equation of this example is a circle. This circle is centered at the origin, and the radius is 7.

y  = 3 - x ²

y  = - a x ² + b x + c

Because a  is negitive parabola opens down ward.

The second equation is an opens down ward parabola. It is shifted vertically 3 units and flipped upside down. Because both of the inequality signs in this example include the equality line underneath (the first one is “less than or equal to” and the second is “less than or equal to”), both lines should be solid.

To determine which  plane to be shaded use a test point in either plane.

A Simple choice is (0,0) substitute x  = 0 and y  = 0 in original inequality

x ² + y ² ≤ 49

0 ≤ 49

The above statement is true .

so you shade inside the circle.

Now plug the same point into the parabola to get

y  = 3 - x ²

0 ≤ 3

The above statement is true .

So you shade inside the parabola.

The solution of the system is the set of ordered pairs in the intersection of the graph of

x ² + y ² ≤ 49 and y  ≤ 3 - x ² . This region is shaded in light purple color.