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When you're looking at a graph, you're often seeing what's called a rectangular coordinate system, which is also known as a Cartesian coordinate system. Imagine it as a grid with two perpendicular lines, one running from left to right (the x-axis) and the other running from top to bottom (the y-axis). These axes intersect at a point called the origin, denoted by coordinates (0, 0).

Each point on the grid is determined by an ordered pair of numbers, written as \( (x, y) \), where \( x \) is the value on the horizontal axis and \( y \) is the value on the vertical axis. For example, the coordinate (3, 4) means you move 3 units to the right along the x-axis and 4 units up along the y-axis. The rectangular coordinate system allows us to plot points, lines, and curves to represent relationships between variables.

A quadratic function is a type of polynomial represented by the equation \( y = ax^2 + bx + c \), where \( a \), \( b \) and \( c \) are constants, and \( a \eq 0 \). It's called 'quadratic' because the highest exponent of the variable x is 2. The graph of a quadratic function is a curve called a parabola. Parabolas have unique shapes and they open either upwards or downwards, depending on the sign of the coefficient \( a \).

As seen in our exercise, when the equation is \( y = \frac{1}{2}x^2 \), this represents a simplified quadratic function where \( a \) is \( \frac{1}{2} \), and \( b \) and \( c \) are both 0. The graph of this function is a parabola that opens upwards since \( \frac{1}{2} \) is positive. As the value of \( x \) increases, the value of \( y \) grows quadratically, meaning it doesn't just double, it squares and is then halved, as indicated by our specific function.

The task of plotting points is fundamental to graphically representing mathematical relationships. Each calculated pair \( (x, y) \) from your equation corresponds to a single point on the graph. To plot a point, you start at the origin (0, 0), move \( x \) units along the x-axis in the direction that corresponds to the sign of \( x \), and then move \( y \) units parallel to the y-axis, again in the direction given by the sign of \( y \).

After computing the \( y \) values in our table using the quadratic function \( y = \frac{1}{2}x^2 \) for each \( x \) value, we get a set of points. For example, with \( x = 2 \) we calculate \( y = 2 \) and plot the point (2, 2) by moving 2 units to the right and 2 units up from the origin. For all our computed points, we follow this process. When connected, these plotted points illustrate the shape of the quadratic function's graph on the rectangular coordinate system.