Graph the following on the same coordinate system. (a) y = x^2 (b) y = 3x^2 (c) y = 1/3x^2 (d) How does the coefficient of x2 affect the shape of the graph?

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the basic shape and properties of parabolas is essential for graphing and analyzing these functions.

The coefficient 'a' in the quadratic function y = ax^2 significantly influences the graph's shape and width. A larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value leads to a wider parabola. This concept is crucial for comparing the graphs of different quadratic functions, as it directly affects their steepness and the rate at which they rise or fall.

Graphing on a coordinate system involves plotting points based on their x and y values to visualize mathematical relationships. For quadratic functions, key points such as the vertex and intercepts are often plotted to accurately represent the curve. Understanding how to effectively use the coordinate plane is vital for comparing multiple functions, as required in this question.