Problem 50 - Algebra and Trigonometry

Question

\(47-50\) . Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$ \left\\{\begin{array}{l}{y \geq x^{3}} \\ {2 x+y \geq 0} \\ {y \leq 2 x+6}\end{array}\right. $$

Step-by-Step Solution

Answer

The vertices of the feasible region are approximately at (0.0, 0.0), (2.0, 2.0), and (-3.0, 0.0).

Step 1: Graphing the Function \(y \geq x^3\)

Open your graphing calculator and input the equation \(y = x^3\). Since the inequality is \(y \geq x^3\), you will shade the region above the curve. This represents all the points where \(y\) is equal to or greater than \(x^3\).

Step 2: Graphing the Line \(2x + y \geq 0\)

Input the equation \(y = -2x\). The inequality \(2x + y \geq 0\) means \(y\) must be greater than or equal to \(-2x\). Shade the area above the line. This shaded area contains all possible solutions to the inequality.

Step 3: Graphing the Line \(y \leq 2x + 6\)

Input the equation \(y = 2x + 6\) into the calculator. For the inequality \(y \leq 2x + 6\), shade below the line on the graph. This represents all solutions where \(y\) is less than or equal to \(2x + 6\).

Step 4: Identifying the Feasible Region

The feasible region, where all three inequalities overlap and are true at the same time, is located where the shaded areas from all three graphs overlap on the calculator.

Step 5: Finding the Vertices of the Feasible Region

Identify the intersection points of the functions and inequalities. These are the coordinates that likely form the vertices of the feasible region. Calculate systematically or use the 'intersect' function of the calculator to find exact points.

Step 6: Rounding the Coordinates

For each vertex found, round the coordinates to one decimal place as required by the problem.

Key Concepts

System of Inequalities

A system of inequalities consists of two or more inequalities with the same set of variables. Their primary goal is to define a region on the graph where all conditions are satisfied simultaneously. For our problem, we have three inequalities:

• First, \(y \geq x^3\), which means for any value of \(x\), \(y\) should be greater than or equal to the cube of \(x\).
• Next, \(2x + y \geq 0\), which suggests that the total of twice \(x\) and \(y\) must not be negative. This is another way of saying \(y\) must be greater than or equal to \(-2x\).
• The last inequality, \(y \leq 2x + 6\), indicates that \(y\) should be less than or equal to \(2x + 6\).

When you graph each inequality, they form boundaries on the xy-plane. The purpose is to find where these shaded areas overlap to identify the feasible region.

Vertices of Feasible Region

The vertices of the feasible region are key points where the boundaries of the shading areas intersect. These points are crucial because they can be potential solutions that satisfy all the inequalities in a system. For our specific problem, after graphing the system of inequalities, the overlapping shaded area or feasible region is located where all individual shaded regions from the ever inequalities meet.

• To find these vertices, you need to look at each pair of equations or inequalities to discover where their lines or curves meet. These are typically "corner" points of the feasible region.
• By using either manual calculation or a graphing calculator's 'intersect' feature, you can pinpoint these intersections more efficiently.
• Once identified, round these intersection coordinates to one decimal place for precision, as directed by the problem.

Graphing Calculator Usage

A graphing calculator is an invaluable tool when dealing with systems of inequalities. It enables visualization of complex solutions and expedites the process of identifying the intersection points or vertices of feasible regions.

To effectively use a graphing calculator for this, follow these steps:

• Input each equation into the calculator section and graph them one by one. As you input inequalities, you'll shadow either above or below the graph based on the inequality direction (greater than or less than).
• Adjust the viewing window as necessary to ensure the overlapping areas are clearly visible.
• Utilize the calculation functions of the calculator, such as 'intersect', to automatically identify points where the graphs intersect, which correspond to the vertices of the feasible region.

Mastering these calculator features will save time and improve accuracy when working with systems of inequalities.