Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&y \geq(x-2)^{2}+3\\\&y \leq-(x-1)^{2}+6\end{aligned}$$

Step-by-Step Solution

Answer

The solution is the overlapping region between the two parabolas on the graph.

Step 1: Analyze the first inequality

The first inequality is \(y \geq (x-2)^2 + 3\). This represents the region above or on the parabola with vertex at point \((2, 3)\). The parabola opens upwards because the coefficient of \(x^2\) is positive.

Step 2: Graph the first parabola

On a graph, plot the vertex \((2, 3)\) and draw a parabola opening upwards. It is a solid line because the inequality includes equality (\(y \geq\)). The region above this curve is part of the solution set of the first inequality.

Step 3: Analyze the second inequality

The second inequality is \(y \leq -(x-1)^2 + 6\). This is a parabola with vertex at point \((1, 6)\) and opens downwards due to the negative coefficient of \(x^2\).

Step 4: Graph the second parabola

Plot the vertex \((1, 6)\) on the same graph, and draw a parabola opening downwards. Use a solid line since the inequality includes equality (\(y \leq\)). The region below this parabola satisfies the second inequality.

Step 5: Identify the solution region

The solution to the system of inequalities is the overlapping region on the graph where both conditions are met. This is the area where the shaded regions of both inequalities intersect.

Key Concepts

Parabolas

Parabolas are a key concept in graphing systems of inequalities because they form the boundary lines for the solution regions. A parabola is a U-shaped curve on the coordinate plane which is defined by a quadratic equation of the form \(y=ax^2+bx+c\). The vertex is the highest or lowest point of the curve, depending on whether it opens upwards or downwards. For example, in the inequality \(y \geq (x-2)^2 + 3\), the parabola opens upward, meaning it looks like a U. This is because the coefficient of the \(x^2\) term is positive.
On the other hand, for \(y \leq -(x-1)^2 + 6\), the parabola opens downward, resembling an inverted U. The negative sign in front of \(x^2\) causes this orientation. Understanding the direction a parabola opens is crucial to correctly shading the solution region in graphing inequalities.

Coordinate Plane

The coordinate plane is the setting where we graph equations and inequalities. It is a two-dimensional surface formed by the intersection of two number lines: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are represented as pairs \((x, y)\).
The first number in the pair is the x-coordinate, representing a position along the horizontal direction, and the second number is the y-coordinate, indicating a position along the vertical direction. When graphing parabolas, like in the exercise, you plot their vertices as points on this plane.

• For \((x-2)^2 + 3\), the vertex is at \((2, 3)\).
• For \(-(x-1)^2 + 6\), the vertex is at \((1, 6)\).

The coordinate plane helps you determine not just the position of these points, but also how the parabolas interact and where the solutions lie. Correctly portraying these relationships visually makes finding the solution sets of systems of inequalities more straightforward.

Solution Sets

Solution sets in graphing systems of inequalities refer to the regions on the graph where all the conditions specified in the inequalities are satisfied simultaneously. When you have more than one inequality, like in this exercise, you're looking for the overlapping regions that satisfy all inequalities involved.
For the inequalities \(y \geq (x-2)^2 + 3\) and \(y \leq -(x-1)^2 + 6\), you need to identify where both shading regions overlap on your coordinate plane. This overlap represents the solution set.
To graphically illustrate the solution set:

• Draw a solid horizontal line for each parabola since both inequalities include equality.
• Shade the area above the upward-opening parabola for the \(\geq\) inequality.
• Shade below the downward-facing parabola for the \(\leq\) inequality.
• The intersection of these shaded regions is your solution set.

By focusing on this overlapping area, you visually interpret which combinations of \(x\) and \(y\) are solutions to the system as a whole.