Problem 57 - A Graphical Approach to Precalculus with Limits

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

Shade the region between the parabolas \( y = (x+2)^2 \) and \( y = -2x^2 \) where they overlap.

Step 1: Understand the Inequalities

The first inequality is a parabola opening upwards, given by \( y \leq (x+2)^2 \). Here, the boundary parabola is \( y = (x+2)^2 \), and the region of interest is below or on this parabola. The second inequality is \( y \geq -2x^2 \), which is a downward-opening parabola, and the region of interest is above or on this parabola.

Step 2: Graph the Boundary Curves

Start by graphing the boundary of each inequality. For \( y = (x+2)^2 \), this is a parabola that shifts the standard \( y = x^2 \) left by 2 units. Plot and draw this parabola on your graph. For \( y = -2x^2 \), this is a wider parabola opening downwards centered at the origin. Draw this parabola as well.

Step 3: Identify the Solution Region

Determine the regions defined by each inequality. The solution to \( y \leq (x+2)^2 \) is the area on or below the parabola. The solution to \( y \geq -2x^2 \) is the area on or above the parabola. The solution set for the system of inequalities is the overlapping region of these two areas. Shade this overlapping region to indicate the solution set.

Key Concepts

Parabolas

A parabola is a curve where any point is at an equal distance from a fixed point, called the focus, and a fixed line, called the directrix. When we talk about graphing parabolas, it’s important to recognize the standard forms:

• Vertical parabolas have the form: \[ y = ax^2 + bx + c \] where they open upwards if \( a > 0 \) and downwards if \( a < 0 \).
• The vertex of the parabola is given by \( (h, k) \) for the equation \[ y = a(x-h)^2 + k \].

In the given exercise, the first inequality, \( y \leq (x+2)^2 \), represents a parabola that opens upwards. This means it will be wide and has its vertex at \( (-2, 0) \).The second inequality, \( y \geq -2x^2 \), defines a parabola opening downwards because of the negative coefficient. Its vertex is at the origin \( (0, 0) \) and it stretches wider than the standard \( y = x^2 \).Understanding these shifts and the direction in which parabolas open helps in predicting the areas they will cover on a graph, which is crucial when solving inequalities.

Inequalities

Inequalities involve relationships where one expression is either less than or greater than another. In our case, we are dealing with quadratic inequalities. Each inequality creates a region on the graph. Let’s break down:

• For \( y \leq (x+2)^2 \),the inequality indicates the shaded region will be located beneath or on the parabola itself. This kind of inequality is known as a non-strict inequality because it includes the boundary.
• For \( y \geq -2x^2 \),the inequality suggests that the region of interest is above or on the parabola. Again, this also includes the boundary line—which is important when graphing both inequalities as the complete boundary must be drawn.

The system of inequalities creates overlapping regions when graphed, which tells us where the solution set lies. Always ensure to graphically represent these inequalities accurately by testing points if needed.

Solution Sets

The solution set for a system of inequalities is the part of the graph where the regions defined by each inequality overlap. In our example, the task involves finding where the two parabolas intersect and ensuring the regions defined by inequalities align.When drawing \( y \leq (x+2)^2 \),the solution set extends downward from the boundary parabola. Conversely, the solution for \( y \geq -2x^2 \)extends upward. The common area, or the solution set, is identified where both conditions satisfy.To represent the solution set:

• Graph each parabola precisely, observing their vertex positions and directions.
• Shade the appropriate regions: below the first parabola and above the second.
• The intersection of these shaded regions represents the solution set, which should be highlighted clearly on your graph.

By shading and overlapping regions, students can visually determine solution sets, providing a clear and interactive way to understand inequality solutions better.