[b]Given a point, like (1, -3), can you find the equations of the tangents to [math]y=x^2[/math] that go through that point?[/b]

1. Show that the tangents to [math]y=x^2[/math] through (-1,1) and (3,9) go through the point (1, -3).[br][br]2. Find the equations of the tangents that go through (2, 3).[br][br]3. Choose a point on the negative y-axis, and find the equations of the tangents that go through that point.[br][br]4. Choose any other point not in the grey region, and calculate the equations of the tangents through that point.[br][br][b]Investigation[/b][br][br]5. What is different about the grey region? Can you prove it?[br][br]6. In questions 1 and 2, the tangent points had 'nice' (integer) coordinates. Can you find other 'through points' which give nice tangent points? Can you find a general rule?[br][br]7. What is the general formula for the equations of the tangents through a point [math](x_1, y_1)[/math]?