Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}y \geq-3 \\ y \leq 1-x^{2}\end{array}\right.$$

Short answer

The solution to the system of inequalities is the area between the line \(y = -3\) and the parabola \(y = 1 - x^{2}\), including the line and the parabola.

Step-by-step solution

Drawing the line \(y = -3\)

The first inequality \(y \geq-3\) represents a horizontal line at \(y = -3\) and all the area above it. For \(y \geq-3\), this includes all points along the line and also extends upwards (all y-values greater than -3). Sketch the horizontal line at \(y = -3\) on the graph and shade the region above this line.

Drawing the parabola \(y = 1 - x^{2}\)

The second inequality \(y \leq 1-x^{2}\) is a parabola opening downwards with its vertex at (0,1). Thus, the solution set for this inequality includes all points along the parabola and extends downwards (where y-values are less than or equal to \(1 - x^{2}\)). Draw this parabola on the same graph and shade the region below this parabola.

Identifying the Solution Area

The solution set for the system of inequalities is the intersection of the solution sets of each individual inequality. Therefore, you need to find the area that is shaded under both inequalities. This will be the space that lies between the line \(y = -3\) and the parabola \(y = 1 - x^{2}\), inclusive.

Key concepts

Graphing Inequalities

When graphing inequalities, the aim is to visually represent all the possible solutions to an inequality on a coordinate plane. This is similar to graphing equations, but instead of a single line or curve, a region is shaded to show the range of possible solutions.

To start, you plot the boundary line or curve of the inequality. For inequalities like ">=" or "<=" the boundary is included in the solution set, which means drawing a solid line or curve. For ">" or "<" the boundary is not included, hence you draw a dashed line or curve.

Next, choose a test point (usually the origin, if it's not on the boundary) to see if it satisfies the inequality. If it does, shade the side of the line or curve containing the test point. If not, shade the opposite side.

• Key steps include determining the boundary type (solid vs dashed), plotting it, and shading the appropriate region.
• Graphing multiple inequalities means finding the overlapping shaded areas to identify the solution set.

Understanding how to translate inequalities into graph regions will help you visualize the feasible solutions.

Parabolas

Parabolas are specific types of curves on a graph, shaped like an open bowl or a U. They are the graphical representations of quadratic functions. A standard parabola can be described by the function \( y = ax^2 + bx + c \). The vertex form, which is useful for graphing, is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

Parabolas can open upwards or downwards. This is determined by the sign of the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

Furthermore, the vertex of the parabola is the highest or lowest point, depending on its orientation.

In the inequality \( y \leq 1-x^2 \), the parabola opens downwards with a vertex at \( (0, 1) \). Knowing these properties helps in shading the correct region when graphing.

Solution Sets

Solution sets in systems of inequalities are the regions where all inequalities in the system overlap. These regions represent all the possible solution points that satisfy each inequality in the system simultaneously.

To find this region, first graph each inequality on the same coordinate plane. Each inequality will have its own shaded region. The solution set is where these shaded areas overlap.

For the given inequalities \( y \geq -3 \) and \( y \leq 1-x^2 \), graphing results in two regions. The solution set is the space that is shaded by both the region above \( y = -3 \) and the region below \( y = 1-x^2 \).

• It’s important to understand that the solution set only includes the intersection of the shaded areas from each inequality.
• Double-checking this overlap ensures that no solutions are missed and the correct region is identified.

By mastering the concept of solution sets, students can effectively determine the feasible range of solutions for systems of inequalities.