Final answer:
Transformations of the function y=x², such as vertical and horizontal shifts, vertical stretching or compressing, and reflections, affect the shape and position of its graph, which is a parabola. Examples include y = x² + 2 for vertical shift and y = -x² for reflection.
Step-by-step explanation:
To describe the transformation from the common function y=x², it is necessary to understand how changes to the function's equation affect its graph. The equation y = x² represents a parabola that opens upwards and has its vertex at the origin (0,0). Transformations can shift, stretch, compress, or reflect this parabola.
For example, the function y = x² + 2 represents a vertical shift upwards by 2 units from the original parabola y = x². If we have a negative sign in front of x, such as in y = -x², it would reflect the parabola across the x-axis, flipping it upside down. Similarly, to stretch or compress the graph vertically, we multiply x by a constant factor. A function like y = 2x² would stretch the graph vertically by a factor of 2, making it narrower than the original parabola.
Other transformations include horizontal shifts and stretches. An equation in the form y = (x - h)² translates the parabola horizontally h units to the right if h is positive, or to the left if h is negative. For stretching or compressing horizontally, an equation like y = x²/c with c > 1 would compress the graph, while c < 1 would stretch it.
To visualize these transformations, one can plot a few data pairs for the transformed function, as given in the example with y = x² + 2. This helps in understanding the effect of the transformation on the graph.