Problem 27 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$\frac{5}{2} y-3 x^{2}-6 \geq 0$$

Step-by-Step Solution

Answer

The graph of the inequality \(\frac{5}{2} y-3 x^{2}-6 \geq 0\) consists of the parabola \(y =\frac{3}{5}2x^{2}+\frac{12}{5}\) and the region above it.

Step 1: Simplify the inequality

Divide every term in the inequality by \(\frac{5}{2}\) to isolate \(y\). This gives \( y \geq \frac{3}{5}2x^{2}+\frac{12}{5}\).

Step 2: Plot the boundary

Next task is to plot this quadratic function \(y =\frac{3}{5}2x^{2}+\frac{12}{5}\). This function forms the boundary of the solution. Start by creating a table of values for x and use it to plot this parabola.

Step 3: Determine the solution region

As the inequality has greater than or equal to symbol, the region above the parabola, including the boundary, is the solution. A point above the parabola satisfies the inequality. Let's choose a test point to confirm. A good choice for the test point is the origin (0,0) as it's easy to compute. If you replace x and y in the inequality with 0, the left side of the inequality is 0 and the right side is \(\frac{12}{5}>0\). Hence, the solution region is correctly below the parabola.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra. They are expressed in the standard form as \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In the context of our problem, the quadratic function was identified as \(y = \frac{3}{5}2x^2 + \frac{12}{5}\). This is a slightly different representation, but aligns with the standard form.Quadratic functions graph as parabolas, which are U-shaped curves. The direction of the parabola (upwards or downwards) depends on the coefficient \(a\). If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards. For our specific function, \(a = \frac{3}{5}2\), a positive value, indicating the parabola opens upwards. Understanding the basic structure of quadratic functions helps in predicting the shape and position of the graph.

Solution Region

The solution region of an inequality represents all the points on a graph that satisfy the given inequality. For the inequality \(\frac{5}{2} y - 3x^2 - 6 \geq 0\), the solution region involves finding which parts of the graph fulfill this condition. In graphing terms, this region can either lie above or below the boundary of the parabola, which is determined by the sign of the inequality. For a \(\geq\) inequality, the solution region will include the area on or above the curve. To find the solution region, one effective strategy is to use test points, such as the origin (0,0). By substituting these coordinates into the inequality, we can check if they satisfy the condition, confirming whether the region falls above or below the boundary.

Boundary

The boundary in graphical terms is the line or curve that illustrates the edge of the solution region. When graphing inequalities, the boundary is often defined by turning the inequality into an equation. For instance, the inequality \(y \geq \frac{3}{5}2x^2 + \frac{12}{5}\) becomes the equation \(y = \frac{3}{5}2x^2 + \frac{12}{5}\) to define the boundary of our solution set.Since this is a quadratic function, the boundary is a parabola. Creating a table of values for x and computing corresponding y-values allows for plotting the parabola on a coordinate plane. Once plotted, it serves as a visual guide to finding which regions satisfy the original inequality. It's important to note that when the inequality is \(\geq\) or \(\leq\), the boundary itself is part of the solution. This means that points on the parabola satisfy the inequality just as well as those in the solution region.