The function `y = x^{2}` is called a quadratic function because the highest degree of its terms is 2.

The standardized form of a quadratic function is `y = ax^{2}+bx+c` , where a, b, and c are real numbers and `a \ne 0` . Note that if a = 0, then we obtain a linear equation. When graphed on the cartesian plane, the graph of this function yields a curve that is called a parabola. A parabola is a u-shaped curve that can open upwards or downwards.

To graph the function `y = x^{2}` on the cartesian plane, we need to obtain the following points:

- Vertex of the parabola: this is the turning point of the curve. It can be a minimum point or a maximum point. If it is a minimum point, then the curve opens upwards. However, if it is a maximum point, the curve opens downwards. The formula for calculating the x-coordinate of the vertex is `x = \frac{-b}{2a}` , where a and b are the coefficients of `x^{2}` and `x` , respectively.

For `y = x^{2}` , a = 1, b=0, and c = 0. Thus, x = `\frac{-0}{2} = 0` .

When x = 0, y = `0^{2} = 0` ; thus, the coordinates of the vertex are (0, 0).

-y-intercept: this is the point where the curve cuts the y-axis. At this point, x = 0. Therefore, in the function `y=x^{2}` , when x = 0, y = `0^{2} = 0` ; thus, the coordinates of the y-intercept are (0, 0).

-x-intercept: this is the point where the curve cuts the x-axis. At this point, y=0. Therefore, when y = 0, then `x^{2} = 0` , and x = 0. Thus, the coordinates of the x-intercept are (0, 0).

- Finally, we can determine the coordinates of a few more points on both sides of the vertex of `y = x^{2}` . We do this as follows:

When x = -2, y = `(-2)^{2} = 4` ; the coordinates of this point are(-2, 4).

When x = -1, `y = (-1)^{2} = 1` ; the coordinates of this point are (-1, 1).

When x = 0, y = 0; (0, 0).

When x = 1, `y = (1)^{2} = 1` ; the coordinates of this point are (1, 1).

When x = 2, `y = (2)^{2} = 4` ; the coordinates of this point are (2, 4).

Having found the points that the parabola passes through, we can now plot them and sketch a parabola through them. Please refer to the attached graph.

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