Grade 11: Parabolas and Lines
1. At the right I have drawn the graph along with a number of secants all centred at x = 1. That is the midpoint of these secants is always at x = 1. By this I mean that the x-intervals defining the secants are all centred at x=1.
What do you notice about these secants? They are parallel.
Is this to be expected? Does it always happen? We use algebra to answer these questions.
2. At the right I have drawn the graph y=x(6-x) along with a number of parallel lines. Some intersect the parabola twice, some not at all, and our graphical intuition tells us that exactly one of the lines intersects only once. We use a parameter b to index this family of lines and find conditions on b for there to be 2, 1 or 0 intersections with the parabola. And thereby we confirm our intuition.
3. At the right I have again drawn the graph y=x(6-x) but now I have drawn a pencil of lines all passing through (5, 5). Some of these lines intersect the parabola twice and others have only one intersection. I ask how many have only the intersection x = 5 and this time I get some variation in the answers given.
Again we parametrize the family and find conditions on the parameter for there to be 2 or 1 intersections with the parabola. And thereby we verify our intuition.