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Quadratic functions are polynomial functions with a degree of 2. This means the variable is raised to the second power as the highest degree. The general form of a quadratic function is given by:\[ y = ax^2 + bx + c \]where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is called a parabola. This can either open upwards, like a cup \( (a > 0) \), or downwards, like a frown \( (a < 0) \).
In the equation from the exercise, \( y = -x^2 + x \), we see immediately that the parabola opens downwards, given that the coefficient of \( x^2 \) is negative. Understanding the shape and orientation of the parabola is crucial when analyzing or sketching quadratic functions, especially when they are part of inequalities.
Quadratic functions are prevalent in various fields of studies, including physics, engineering, and economics, because they describe the motion of objects, the shape of certain structures, and other natural phenomena.

The coordinate plane, often called the Cartesian plane, is a two-dimensional plane formed by two perpendicular lines called axes. These axes divide the plane into four quadrants.
The x-axis is the horizontal line, and the y-axis is the vertical line. The point where these two axes intersect is termed as the origin, denoted by \((0, 0)\). Each point on the coordinate plane is identified using an ordered pair \((x, y)\).
When plotting the quadratic function \( y = -x^2 + x \), it's vital to understand how each point changes with respect to both axes. The equation divides the plane into regions where different solutions to the inequality can be determined. The space above, below, or on the curve needs to be inspected to identify where the inequality holds true.
Visualization on the coordinate plane helps us see the relationship between the function and its inequality. It is a powerful tool for solving and understanding algebraic equations and inequalities.

The vertex of a parabola is a crucial point that describes its maximum or minimum position. For a parabola given by \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula:\[ x = -\frac{b}{2a} \]Once the x-coordinate is computed, substitute it back into the original equation to find the y-coordinate of the vertex.
In the given function \( y = -x^2 + x \), the vertex is calculated as:\[ x = -\frac{1}{2(-1)} = 0.5 \]Plugging \( x = 0.5 \) back into the equation gives:\[ y = -0.5^2 + 0.5 = 0.25 \]So, the vertex of this parabola is \((0.5, 0.25)\). This point marks the highest point of the parabola because it opens downwards. Understanding the vertex's location helps in sketching the graph and analyzing which side of the parabola should be shaded to represent the solution set for the inequality, \( y < -x^2 + x \).