Describe in words the graph of each of these curves below. Include in your description the shape, along with other possible relevant information such as length, width, and center points.

a. Y = 3X2

b. (X-1)2 + (Y-8)2 = 16

c. (X+2)2 + (Y-4)2 = 36

d. Y = X2 + X

a. The graph of y = 3x^2 is a parabola facing upwards. It has a symmetric shape, with the vertex at the origin (0, 0). The length of the parabola extends infinitely in the positive and negative x-directions, while the width of the parabola gradually widens as the x-values increase.

b. The equation (x-1)^2 + (y-8)^2 = 16 represents a circle. The center of the circle is at point (1, 8) and the radius is 4. The shape is perfectly round, with a constant width throughout.

c. The equation (x+2)^2 + (y-4)^2 = 36 represents another circle. The center of the circle is at point (-2, 4) and the radius is 6. Similar to the previous example, the graph is round and maintains the same width at all points.

d. The equation y = x^2 + x represents a parabola facing upwards. The shape is similar to the one in example (a), but shifted horizontally. The vertex of the parabola is at the origin (0, 0), and it extends infinitely in the positive and negative x-directions. The width of the parabola gradually increases as the x-values increase.