Problem 48 - Precalculus with Limits

Question

In Exercises 47-52, use a graphing utility to graph the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{y<-x^{2}+2 x+3} \\ {y>x^{2}-4 x+3}\end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the system of inequalities is composed of two parabolas. The solution set of the system of inequalities is the region of intersection between the shading beneath the downward-opening parabola and above the upward-opening parabola.

Step 1: Analyze Inequalities

In the system of inequalities, there are two inequalities: \(y<-x^{2}+2 x+3\) and \(y>x^{2}-4 x+3\). These are quadratic equations, so their graphs will be parabolas. The first inequality's graph opens downwards, because the coefficient of \(x^2\) is negative while the second inequality's graph opens upwards, because the coefficient of \(x^2\) is positive.

Step 2: Graph the Inequalities

Next, each of the inequalities will be graphed onto the same graph, using a graphing utility. The areas under the graph of the first inequality and above the graph of the second inequality will both be shaded. This shading signifies the set of all \(y\) values which satisfy the inequality for a given \(x\) value.

Step 3: Define the Solution

The solution to the system of inequalities is the intersection of the shaded regions from both inequalities. This will be the area that is shaded both above and below, satisfying both \(y<-x^{2}+2 x+3\) and \(y>x^{2}-4 x+3\).

Key Concepts

Graphing Parabolas

Understanding how to graph parabolas is a crucial step in solving quadratic inequalities. In our exercise, the parabolas are represented by the quadratic expressions in the inequalities. Here are some key points to remember when graphing parabolas:

• A parabola is the graph of a quadratic function, which typically takes the form \( y = ax^2 + bx + c \).
• Determine the direction in which the parabola opens by looking at the coefficient of \( x^2 \). If it is positive, the parabola opens upwards; if negative, it opens downwards.
• The vertex, which is a specific point where the graph changes direction, is an essential feature of a parabola. It can be found using the formula \( x = -\frac{b}{2a} \).

When graphing the parabola for \( y < -x^2 + 2x + 3 \), you should first plot the boundary parabola, \( y = -x^2 + 2x + 3 \), which opens downwards. Similarly, for \( y > x^2 - 4x + 3 \), plot the boundary \( y = x^2 - 4x + 3 \), which opens upwards. Remember, the solution set is determined by the areas that satisfy the inequalities, which we'll cover in the next sections.

Solution Set

The solution set of a system of inequalities refers to all the points that satisfy each inequality involved. In our system, it's important to recognize the combined effects of both inequalities.

• The boundary lines, \( y = -x^2 + 2x + 3 \) and \( y = x^2 - 4x + 3 \), are crucial in defining these regions. However, only the points that truly satisfy the \( < \) and \( > \) conditions are part of the solution set.
• Graphically, the solution set is the intersection area, where the shaded regions of each inequality overlap.

To identify the solution set:
1. Shade the region below the line \( y = -x^2 + 2x + 3 \) because the inequality specifies \( y < -x^2 + 2x + 3 \).
2. Shade the region above the line \( y = x^2 - 4x + 3 \) as the inequality states \( y > x^2 - 4x + 3 \).This overlapping shaded area on your graph is the solution set, where both inequalities are true.

System of Inequalities

A system of inequalities involves multiple inequalities that are solved together. Solving or graphing these systems helps us understand where the inequalities intersect in terms of solution sets.

• The key task is to graph each inequality carefully and look for overlapping regions which represent solutions that satisfy all inequalities.
• Intersection points and the overlapping areas of the shaded regions form the solution set.

In the current exercise, we deal with quadratic inequalities forming parabolic boundaries. This makes the task more visually intensive as you plot these curves and determine where they meet the given conditions. Working with systems of inequalities often involves trial and error or the use of graphing utilities to verify the solution areas correctly. Once the graphs are plotted and shaded appropriately, the intersection of the shaded regions gives us the solution to the system, showing which \( x, y \) pairs are viable solutions according to both inequalities.