Problem 32 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the inequality. $$ y<-x^{2}+x $$

Step-by-Step Solution

Answer

In simple terms, the solution to the inequality \(y < -x^{2} + x \) is the region below the graph of the function \(f(x) = -x^2 + x\) excluding the points on the parabola.

Step 1: Rewrite inequality

Firstly, let's rewrite inequality for better interpretation. Rewrite inequality as \(y + x^2 - x < 0\).

Step 2: Sketch the quadratic function

We then sketch the function \(f(x) = -x^2 + x\). This is a downward-opening parabola with vertex at \((0.5, 0.25)\) and axis of symmetry \(x = 0.5\).

Step 3: Identify the region

The inequality is \(y < -x^2 + x\) which means \(y\) values are less than those given by the quadratic equation. It represents the region below the graph of the equation. We shade this area.

Step 4: Determine the solution

Any point in the shaded area or on the parabola represents a solution to the given inequality. Omit the points on the parabola as the inequality is strict (<)

Key Concepts

Quadratic Functions

Quadratic functions form the foundation of various algebraic and geometric concepts. They are expressed in the standard form as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, it is rearranged to \(y = -x^2 + x\), indicating a specific type of quadratic function. Quadratic functions have a characteristic curve called the parabola. When graphed, these functions produce a "U"-shaped curve. The direction of this curve depends on the leading coefficient \(a\). If \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards. Quadratic functions are important as they can model various real-world scenarios, like projectile motion or area problems.

Parabolas

Parabolas are the graphical representation of quadratic functions. They have several key traits:

• The vertex, which is the parabola's highest or lowest point depending on its orientation.
• The axis of symmetry, a vertical line passing through the vertex, dividing the parabola into mirror-image halves.
• The direction of opening, which signals whether the parabola opens upwards or downwards. It's determined by the sign of the coefficient \(a\) in the equation.

In our inequality example \(y < -x^2 + x\), the parabola opens downwards because the quadratic term is negative. Observing parabolas can provide insights into the solutions of quadratic inequalities and equations. Practical uses of parabolas include optics and engineering fields where they help in designing reflectors and antennas.

Shading Regions

Shading regions is a method used in graphing inequalities. It helps identify the set of solutions that satisfy the inequality conditions. For the inequality \(y < -x^2 + x\), we first graph the parabola and then determine which side of the parabola contains solutions that make the inequality true.

In this case, since the inequality is "less than", we shade the region below the parabola. This step visually depicts all \(y\) coordinates that fall beneath the curve as solutions. Points on the parabola itself are not included in the solution set because of the strict inequality. Shading makes it easy to identify and communicate which regions satisfy the problem’s conditions. This method is integral in expressing solutions graphically and comprehensively.

Vertex of a Parabola

The vertex is a pivotal aspect of a parabola, providing significant information about the quadratic function. In our specific function \(-x^2 + x\), the vertex is found through various methods like completing the square or applying the vertex formula \(x = -\frac{b}{2a}\). For this equation, the vertex is at \((0.5, 0.25)\).

The vertex represents the maximum height of a downward-opening parabola or the minimum depth for an upward-opening one. It is a crucial indicator of the parabola's shape and orientation. The vertex also serves as a reference point for symmetry, aiding in various mathematical tasks like factoring and optimizing functions. Understanding the vertex and its implications is vital for solving and graphing quadratic equations and inequalities effectively.