Question

Graph each system of inequalities. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 16 \\\y \geq x^{2}-4\end{array}\right.$$

Short answer

The solution is the intersecting region inside the circle and above the parabola.

Step-by-step solution

Identify and Graph the Circle

The inequality \( x^2 + y^2 \leq 16 \) represents a circle with a radius of 4 centered at the origin (0,0). To graph this, draw a circle with a center at (0,0) and a radius extending out to 4 units in all directions. Shade the interior of the circle to indicate all points inside and on the circle satisfy the inequality.

Identify and Graph the Parabola

The inequality \( y \geq x^2 - 4 \) represents a parabola opening upwards with its vertex at (0, -4). To graph this, draw the parabola starting from the vertex (0, -4) and curving upwards. Shade the region above and including the parabola to indicate all points that satisfy the inequality.

Find the Intersection of Regions

Combine the shaded regions from both inequalities. The solution to the system of inequalities is the region where the shaded areas of the circle and the parabola overlap. Shade this intersecting region to represent the final solution.

Key concepts

systems of inequalities

When dealing with systems of inequalities, the goal is to find the region that satisfies all given inequalities. Unlike solving regular equations where you find specific points, with inequalities you are looking for shaded regions that represent a set of possible solutions.
The process involves:

• Graphing each inequality on the same coordinate plane
• Identifying the regions that satisfy each inequality
• Finding the overlapping region where all inequalities intersect

This overlapping region is the solution. Understanding how to graph each inequality individually before combining them is crucial.

graphing circles

Graphing circles involves understanding the general form of the circle's equation, which is (x - h)^2 + (y - k)^2 = r^2 . Here, (h, k) is the center of the circle, and r is the radius.
::In the inequality form, such as x^2 + y^2 ≤ 16, the steps are:

• Identify the center and radius from the equation. In this example, the center is (0, 0), and the radius is √16 = 4.
• Draw the circle with the identified center and radius.
• Shade the region inside the circle if the inequality is ≤ (less than or equal to) or outside if it is ≥ (greater than or equal to).

The circle's boundary is included in the solution if the inequality is ≤ or ≥.

graphing parabolas

Graphing a parabola involves recognizing its standard form: y = ax^2 + bx + c . For inequalities, the steps involve similar graphing as for equations but include shading:

• Identify the vertex of the parabola. For instance, in y ≥ x^2 - 4, the vertex is at (0, -4).
• Determine the direction it opens (upwards if 'a' is positive, downwards if 'a' is negative).
• Draw the parabola accordingly.
• Shade the region above the parabola if the inequality is ≥ or below if it is ≤.

Remember to include the boundary line as part of the solution if the inequality is ≥ or ≤.

inequalities intersection

Finding the intersection of inequalities is like finding a common area where all conditions are met:

• Graph each inequality separately on the same coordinate plane.
• Each inequality will have a shaded region representing its solution.
• The final solution is where the shaded regions overlap.

This intersection is the area that satisfies all inequalities in the system.
For example, combining a circle x^2 + y^2 ≤ 16 with a parabola y ≥ x^2 - 4, the solution region is the overlapping part inside or on the circle and above or on the parabola. This graphical method ensures that all given conditions are met simultaneously.