Problem 44 - Precalculus: Mathematical for Calculus

Question

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, correct 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

Vertices are approximately (0, 0), (-2.5, -8.25), and (-1.5, 3).

Step 1: Graph the First Inequality

The first inequality is \(y \geq x^3\). This means that the region of the graph will include the area above or on the curve of \(y = x^3\). To graph this, plot the curve of \(y = x^3\) and shade the area above it.

Step 2: Graph the Second Inequality

The second inequality is \(2x + y \geq 0\). Rearrange it to \(y \geq -2x\). This describes a line that passes through the origin with a slope of \(-2\). Plot this line and shade the region above the line.

Step 3: Graph the Third Inequality

The third inequality is \(y \leq 2x + 6\). This can be rewritten as \(y \leq 2x + 6\) and represents a line with a slope of \(2\) and a y-intercept at \(6\). Plot this line and shade the region below it.

Step 4: Identify the Intersection Region

The solution to the system of inequalities is the region where the shaded areas from all three inequalities overlap. Use a graph or graphing calculator to precisely identify this area.

Step 5: Determine the Vertices of the Intersection Region

To find the vertices of the overlapped region, solve the system where these curves intersect each other pairwise. 1. **Intersection of the lines \(y = x^3\) and \(y = -2x\):** Solving for \(x^3 = -2x\), factor to get \(x(x^2 + 2) = 0\). Thus, \(x = 0\), as other solutions derive from imaginary numbers, find corresponding \(y\), \(y = 0\). Hence the vertex is (0, 0).2. **Intersection of the lines \(y = x^3\) and \(y = 2x + 6\):** Setting \(x^3 = 2x + 6\), solve numerically or graphically to find intersection point. Approximate solution yields: \(x \approx -2.5\) and \(y \approx -8.25\) results in vertex (approximately) (-2.5, -8.25).3. **Intersection of the lines \(y = -2x\) and \(y = 2x + 6\):** Solving for \(-2x = 2x + 6\), gives \(x = -1.5\), \(y = 3\); vertex (-1.5, 3).

Step 6: Compile the List of Vertices

From the above calculations, the vertices of the intersection region are approximately at the points (0, 0), (-2.5, -8.25), and (-1.5, 3). These points outline the boundary of the shaded overlapping region.

Key Concepts

Graphing Inequalities

Graphing inequalities is an essential skill in understanding and solving systems of inequalities. Each inequality represents a region on a graph, which shares a solution with other inequalities only in the overlapping areas. For instance:

• The inequality \( y \geq x^3 \) implies shading above the cubic curve \( y = x^3 \) on the graph.


• The inequality \( 2x + y \geq 0 \) or equivalently \( y \geq -2x \) involves shading above a line passing through the origin with a descending slope of \(-2\).


• For \( y \leq 2x + 6 \), you shade beneath a line with a slope of \(2\) and a y-intercept of \(6\).

It's crucial to accurately shade all relevant regions so the solutions are correctly represented. Where these shaded regions overlap is key, as it shows the solution to the system of inequalities.

Intersection Points

Finding intersection points between inequalities is about determining exact points where equations meet. These points are usually vertices of the solution region. To find intersections, solve the inequalities pairwise:

• For \( y = x^3 \) and \( y = -2x \), solve \( x^3 = -2x \) to find the intersection point at \((0, 0)\).
• For \( y = x^3 \) and \( y = 2x + 6 \), solve \( x^3 = 2x + 6 \). This might require approximation techniques or graphing to find a point near \((-2.5, -8.25)\).
• For \( y = -2x \) and \( y = 2x + 6 \), solve \(-2x = 2x + 6\), yielding \((-1.5, 3)\).

Identify these intersection points to outline the boundary of the feasible region which satisfies all inequalities. Knowing these vertices helps in understanding and visualizing the feasible solution area in a system of inequalities.

Graphing Calculator Usage

Using a graphing calculator can significantly ease the task of solving inequalities. Here’s how it helps:

• Input each inequality separately, using the calculator's menu or command functions. This will allow you to graph the curves and lines accurately.


• The calculator will typically shade the appropriate region for each inequality, automatically highlighting intersections without manual shading.


• Graphing calculators can numerically identify intersection points, offering precise decimal coordinates for vertices, which are essential when solutions include decimals.

With practice, graphing calculators can become invaluable tools in understanding graphing concepts and verifying hand-drawn graphs. They provide visual clarity and can double-check calculations, especially for complex systems of inequalities.