Problem 45 - A Graphical Approach to College Algebra

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \leq(x+2)^{2}\\\ &y \geq-2 x^{2} \end{aligned}$$

Step-by-Step Solution

Answer

Graph the inequalities, then find the overlapping region, bounded by the curves.

Step 1: Understand each inequality

The first inequality is \( y \leq (x+2)^2 \), which represents the region below or on the parabola with vertex at (-2, 0) opening upwards. The second inequality is \( y \geq -2x^2 \), which represents the region above or on the parabola with vertex at (0, 0) opening downwards.

Step 2: Graph the first inequality

Start by graphing \( y = (x+2)^2 \). This parabola opens upwards with its vertex at (-2,0). Shade the region below this parabola because \( y \leq (x+2)^2 \) indicates this portion.

Step 3: Graph the second inequality

Graph \( y = -2x^2 \). The vertex of this downward opening parabola is at (0,0). Shade the region above this parabola because \( y \geq -2x^2 \) indicates this portion.

Step 4: Identify the solution set

Locate the overlapping shaded region from the two previous steps. This area represents the solution set to the system of inequalities as it satisfies both conditions \( y \leq (x+2)^2 \) and \( y \geq -2x^2 \).

Key Concepts

System of Inequalities

A system of inequalities involves more than one inequality to be solved simultaneously. Each inequality restricts the solution to a certain region of the graph. When you graph these inequalities, you're looking for the overlapping region that satisfies all inequalities in the system. This is because every point in the overlapping region will fulfill all the given conditions.

To solve a system of inequalities, follow these easy steps:

• Start by considering each inequality separately.
• Graph each inequality on the same set of axes.
• Shade the regions that represent the solution of each inequality.
• Finally, identify the overlapping region where all shaded areas meet; this is your solution set.

In this exercise, you’ll apply these steps to inequalities involving quadratic functions, where the solution set is the common shaded region on a graph.

Quadratic Functions

Quadratic functions are equations of the form \( y = ax^2 + bx + c \), which graph as parabolic curves on a coordinate plane. They have distinct features that make them easy to recognize and graph.

The vertex of the parabola, which is the highest or lowest point of the curve, is crucial for graphing and solving inequalities involving these functions. For example, our exercise includes the quadratics \( y = (x+2)^2 \) with vertex at (-2, 0) and \( y = -2x^2 \) with vertex at (0, 0).

Parabolas can open either upwards or downwards:

• If \( a > 0 \), the parabola opens upwards, like a smiley face.
• If \( a < 0 \), the parabola opens downwards, like a frown.

Understanding the direction the parabola opens is key to shading the correct region when graphing inequalities.

Parabolas

Parabolas are U-shaped graphs resulting from quadratic functions and play a big role in graphing inequalities. They can open upward or downward, depending on the coefficient of the \( x^2 \) term.

Each parabola has a vertex and an axis of symmetry.

• The vertex is the point where the parabola changes direction, defined by the formula \( (h, k) \) when in vertex form \( y = a(x-h)^2 + k \).
• The axis of symmetry is a vertical line through the vertex that divides the parabola into two mirror-image halves. It has the equation \( x = h \).

Knowing these parameters helps in sketching the parabola accurately.

In inequality problems, understanding the shape and orientation of a parabola helps determine which region to shade, either inside or outside the curved path, to denote where the inequality holds true.