Question

THOUGHT PROVOKING Draw a company logo that is created by the intersection of two quadratic inequalities. Justify your answer.

Short answer

The chosen inequalities \(y \leq x^2 + 1\) and \(y \geq -x^2 + 1\) intersect to form a diamond-like shape when graphed. Since many companies use geometric shapes in their logos, this intersecting region can serve as a company logo.

Step-by-step solution

Identify suitable quadratic inequalities

First, you are to identify two quadratic inequalities that would intersect to form a shape that draws a company logo when graphed. For example, we may consider the inequalities \(y \leq x^2 + 1\) and \(y \geq -x^2 + 1\). These two inequalities are parabolic and have their vertices at points (0,1). The first inequality opens upwards while the second opens downwards, and when graphed and shaded according to the inequality sign, they intersect to form a diamond shape, which could represent a company logo.

Graph the inequalities

Following the creation of the inequalities, you are to plot these in a graph. The first equation \(y \leq x^2 + 1\) forms a parabola that opens upwards. The inequality implies that the shading should be below the parabola. The second equation \(y \geq -x^2 + 1\) also forms a parabola, but this one opens downwards. Its inequality implies that the shading should be above the parabola. When both inequalities are graphed together on the same set of axes, the intersecting/shaded region forms a diamond-like shape.

Justify your answer

Now you should provide a justification for the chosen inequalities and resultant logo. Explain how the chosen inequalities intersect to form a shape that can be used as a company logo. Due to their intersection, the overlapping shaded region forms a diamond shape. Many companies use geometric shapes, like a diamond, in their logos. Therefore, this intersection of quadratic inequalities could function as a company logo.

Key concepts

Graphing Inequalities

Understanding how to graph inequalities is essential when solving quadratic inequality problems. Graphing an inequality means plotting the region of the coordinate plane that satisfies the inequality. Let's break down the process.

For quadratic inequalities like the ones in the exercise—namely, the inequalities \(y \leq x^2 + 1\) and \(y \geq -x^2 + 1\)—we first graph the related quadratic equations \(y = x^2 + 1\) and \(y = -x^2 + 1\). These graphs are parabolas, which provide the boundary for the inequality. Once the parabolas are drawn, we determine the shading based on the inequality sign. For \(y \leq ...\), we shade below the parabola, and for \(y \geq ...\) above it.

By crossing the shaded regions of both inequalities, we define the intersection area, which represents all the coordinate points that satisfy both inequalities. This intersection, particularly in the given exercise, creates a distinct shape that could serve as a company logo.

Parabolas

The Shape of Solutions: Parabolas

Parabolas are ubiquitous in algebra and appear frequently when working with quadratic equations. A parabola is a U-shaped curve that can open upwards or downwards, determined by the positive or negative coefficient of the \(x^2\) term.

When graphing a parabola for the inequality \(y \leq x^2 + 1\), we note that the vertex is at (0, 1) and the parabola opens upwards. Conversely, with \(y \geq -x^2 + 1\), the parabola opens downwards, also with the vertex at (0, 1). The point where both parabolas intersect creates the boundary of our logo's shape.

The significance of parabolas in graphical solutions is that they provide a visual representation of possible solutions to quadratic inequalities. By visualizing these shapes, students can better understand where the solutions lie in relation to the parabolas.

Algebraic Concepts

Building Blocks: Algebraic Concepts

Algebraic concepts are the foundation of understanding and solving mathematical problems, particularly inequalities. Quadratic inequalities are expressions where the highest exponent of the variable is two. These types of inequalities can take various forms, such as \(ax^2 + bx + c \leq 0\) or \(ax^2 + bx + c \geq 0\), where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

The process of solving these inequalities usually involves finding the roots of the associated quadratic equation and then testing intervals to determine where the inequality holds true. This is done by creating a sign chart or by graphing.

Understanding the graph of a quadratic equation can significantly simplify the process of solving the inequality. Since the graph of a quadratic equation is always a parabola, grasping the shape and position of the parabola can help students identify the solution set for the inequality.