Problem 42 - A Graphical Approach to College Algebra

Question

Checking Analytic Skills Graph the solution set of each system of inequalities by hand. Do not use a calculator. $$\begin{aligned}&y \leq-x^{2}\\\&y \geq x^{2}-6\end{aligned}$$

Step-by-Step Solution

Answer

Graph and shade the overlapping region between the parabolas \(y=-x^2\) and \(y=x^2-6\).

Step 1: Understand the inequalities

We have two inequalities: 1. \( y \leq -x^2 \), which represents the region of the plane below or on the parabola \( y = -x^2 \).2. \( y \geq x^2 - 6 \), which represents the region of the plane above or on the parabola \( y = x^2 - 6 \).

Step 2: Graph the boundary of the first inequality

The boundary of \( y \leq -x^2 \) is the parabola \( y = -x^2 \). This is a downward-opening parabola with its vertex at the origin (0,0). Since the inequality is \( \leq \), the parabola should be graphed using a solid line to indicate points on the parabola are included in the solution.

Step 3: Shade the region for the first inequality

Shade the region below the parabola \( y = -x^2 \). This includes the area inside the curve and below it. This represents all the points where \( y \leq -x^2 \).

Step 4: Graph the boundary of the second inequality

The boundary of \( y \geq x^2 - 6 \) is the parabola \( y = x^2 - 6 \). This parabola opens upwards with its vertex at (0, -6). Since the inequality is \( \geq \), graph it with a solid line to indicate that points on the parabola are included in the solution.

Step 5: Shade the region for the second inequality

Shade the region above the parabola \( y = x^2 - 6 \). This includes the area inside the curve and above it. This represents all the points where \( y \geq x^2 - 6 \).

Step 6: Identify the solution region

The solution set to the system consists of all points that are in both shaded regions simultaneously, i.e., the overlap of the regions shaded in Steps 3 and 5. Visually, this region looks like a 'lens' shape between the two overlapping parabolas.

Key Concepts

Graphing Inequalities

Graphing inequalities can be a breeze once you get the hang of it! The key is to first understand that inequalities describe a range of values.

• When you graph an inequality like \( y \leq -x^2 \), you're looking for all the points below or on the parabola formed by \( y = -x^2 \).
• For \( y \geq x^2 - 6 \), the graph includes points above or on the parabola \( y = x^2 - 6 \).

Start by sketching each boundary equation as if it were just an ordinary function. Since both inequalities involve 'or equal to' (\( \leq \) or \( \geq \)), use a solid line to represent these boundaries. Then, shade the appropriate region. Below the curve for \( y \leq -x^2 \) and above the curve for \( y \geq x^2 - 6 \). This shading helps in visualizing which parts of the graph satisfy the inequality, paving the way to identify the solution set.

Parabolas

Parabolas are intriguing shapes that appear all over in algebra! In our system of inequalities, each parabola behaves differently because of how they open. The equation \( y = -x^2 \) describes a downward-opening parabola. This happens because the coefficient of \( x^2 \) is negative.

• The vertex, the highest point here, is at (0, 0).
• It symmetrically stretches outwards as it descends.

On the flip side, \( y = x^2 - 6 \) forms an upward-opening parabola due to its positive \( x^2 \) coefficient.

• The vertex here is at (0, -6), indicating it starts a little lower in the plane.
• It symmetrically rises upward.

Understanding these properties helps to graph them accurately and realize which side of the parabola is relevant for the inequality solution.

Solution Set

The solution set is like the answer to your puzzle, piecing together the regions that satisfy both inequalities. In this case, you've shaded different parts of the graph for \( y \leq -x^2 \) and \( y \geq x^2 - 6 \). You are now on the lookout for where these shaded areas overlap. The region of overlap is the magical "lens" shape that serves as the solution set.

• It contains all the points that fulfill both conditions at once.
• This overlapping zone gives you all the combinations of (x, y) that solve each inequality.

Keep in mind, only the overlapping area’s points are solutions to the system of inequalities. Understanding this real estate of solutions is crucial for solving complex inequality systems effectively.