Problem 26 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The inequality \(2 x^{2}-y-3>0\) when graphed results in a parabola opening upwards with a dashed line, with the shaded region underneath, excluding points on the line.

Step 1: Re-arrange the equation

Rearrange the inequality into the y = mx + b format, which is more familiar when it comes to graphing. This gives us \( y < 2x^{2}-3\).

Step 2: Graph the Quadratic Inequality

First, draw the parabola \(y = 2x^{2} - 3\). This will be the boundary of the solutions you are looking for. In this case, since the inequality is 'less than' and not 'less than or equal to', the line itself is not included in the solution. Therefore, the line should be a dashed line indicating that the points on the line are not included in the solution.

Step 3: Determine which side of the line to shade

The inequality \(y < 2x^{2} - 3\) tells us that the y-values we're interested in are 'less than' our parabola. This means we should shade the area underneath the parabola.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are mathematical expressions that involve a quadratic function, usually in the forms of either \( f(x) > 0 \), \( f(x) < 0 \), \( f(x) \geq 0 \), or \( f(x) \leq 0 \). These inequalities compare a quadratic equation to zero, and interpreting them often involves determining which sets of x-values satisfy the condition. Quadratic inequalities are typically written in the standard form \( ax^2 + bx + c \) and are solved by either factoring, completing the square, or using the quadratic formula, followed by checking the intervals of interest.
Most importantly, solving these involves not just finding the points where the quadratic is equal to zero (the roots), but also determining if it is greater than or less than zero within intervals bounded by these roots. In graphing, this means shading regions of the graph that satisfy the inequality instead of highlighting individual points.
For example, when graphing \( y < 2x^2 - 3 \), we look for all \( (x, y) \) pairs below the parabola \( y = 2x^2 - 3 \). This approach requires a good understanding of how quadratic equations form parabolas.

Parabolas

Parabolas are the U-shaped graphs formed by quadratic functions, like \( y = ax^2 + bx + c \). When it comes to graphing, the value of \( a \) determines the direction of the parabola:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Parabolas are symmetric around their vertex, which is the peak or the lowest point of the graph. The vertex can be found using the formula \( x = -\frac{b}{2a} \), and once found, you can calculate the corresponding y-coordinate by substituting back into the quadratic equation.
The roots or x-intercepts are where the parabola crosses the x-axis; these points are found by setting \( y = 0 \) and solving the quadratic equation. The y-intercept, on the other hand, is simply the constant term when \( x = 0 \). Understanding these properties helps you successfully graph parabolas and determine the relevant areas that satisfy given inequalities.

Graphing Utility

Graphing utilities are tools (either software or calculators) that assist in visualizing functions and inequalities on a coordinate plane. They are particularly helpful for quadratic inequalities because they allow you to view the entire parabola and identify which regions are part of the solution. These tools make it easy to toggle between plotting equations and inequalities.
When using a graphing utility, begin by entering the quadratic equation, such as \( y = 2x^2 - 3 \), to outline the border of the inequality with either a solid or dashed line, depending on whether the inequality is inclusive (\( \leq \), \( \geq \)) or not (\( < \), \( > \)). For \( y < 2x^2 - 3 \), you'd use a dashed line.
Next, the graphing utility highlights where the inequality holds by shading the region that includes solutions. This lets you quickly confirm areas satisfying the inequality without manually testing individual points, allowing you to see how changes in the equation affect the graph instantly. Understanding how to adeptly use these tools can greatly improve your grasp of functions and inequalities.