Still not done…
The question is not whether a horizontal tangent goes through the vertex but rather whether the tangent through the vertex has to be horizontal. It does, but if you have not had calculus (and you don’t think of the symmetry argument) then it takes some thinking to show that any line through the vertex with a non-zero slope will in fact meet the parabola a second time.
I still think that this is an oddball of an SAT question in that it is so much easier for those who have learned more advanced math (and not just because they are better at math).
No, we answer the question, “If a line is tangent to the vertex, what is one possible equation of the line?”, not “If the line is tangent to the vertex, what is the only possible the equation of the line?”
It is good enough to show that a horizontal line that goes through the vertex is tangent. This is a multiple choice question. You find a line that intersects only the vertex, and then find a point among the answer choices that would be on that line. You don’t have to prove that the line you found is the only line that intersects only the vertex. Who cares if there are other lines. Your answer choice is still correct.
Of course, you COULD prove it algebraically by setting up the system
y=(x-c)^2+d equation of parabola (I assumed for simplicity that a=1)
y-d=m(x-c) equation of line containing (c,d)
Then equate
(x-c)^2=m(x-c) This gives a quadratic in x with constants m and c:
x^2-(2c+m)+c^2+mc=0
You find the discriminant of this quadratic and set it equal to 0 because there is only 1 solution to the system.
(2c+m)^2=4(c^2+mc)
m^2=0
m=0
This method is good for tangents to points other than the vertex.
Notice that here too, the key concept to tangency is that there is exactly one point of intersection.
I have students who have to find tangent lines to conics this way all the time. I believe their school calls this GEOMETRY that uses algebra. They do it at least a year before pre-calc. Obviously much too time-consuming for the SAT.
Isn’t this an rSAT question? There are A LOT of these on the rSat. The kids who have studied physics, math level 2, descriptive statistics are going to be laughing. All those projectile motion questions – ridiculous for people in AP Physics, very hard for people who have never studied physics. rSAT math like a subject test + verbiage.
You are right about the rSAT. But I think this question was from June 2014.
In any case, most of our discussion has just been for fun. On the SAT, for students who did not already know how to do this, I would recommend something much more mindless:
Draw a neat parabola with its vertex at some place other than the origin. So essentially, you are making up numbers for c and d. (When you do that, it’s a good idea to avoid zero.) For example, I put my vertex at (3,2). Then draw your best guess at what a tangent line at the vertex looks like. Hopefully. most students will draw it as horizontal or nearly so. Then go to the answers and rule out anything that is not even close to being on your line.
@pckeller
Yes, I like that method. But if we are talking about the OLD SAT, I would have the student whip out his
TI-Nspire CX CAS
tangentLine(expr,var, val) returns the equation of the tangent line to the function at the given point.
For example,
tangentLine ((x-c)^2+d,x,c)) Enter
returns y=d
Those things are great…
Get one for the calc section of the rSAT. Looks like there are going to be tangent lines there too…