I looked at PWN and someone’s explanation in college confidential but still don’t get it? Why does the line l has slope of 0? (Thanks in advance)
the graph of a function in the xy plane is a parabola that opens upward and has its vertex at point (c,d) if the line l is tangent to the parabola at its vertex, which of the following must be another point on line l?
(-5,d)
(-5,-d)
(0,0)
(c,-5)
(-5,-c)
You raise an interesting question. I believe that this is a real SAT question, yes? And yet it draws on an idea that comes from calculus, at least conceptually. I guess I have to ask: can you picture a tangent line? Think of it as a ski, riding along the curve. The ski just touches the curve at one point, but doesn’t cross it (except at “inflection points” but don’t worry about that for now).
The parabola opens upward. So as you are skiing along the parabola, starting from left of the vertex, your ski points downhill. We say that the tangent line has a negative slope. After you pass the vertex, your ski will be sloping upward. But, AS YOU PASS THRU THE VERTEX, your ski will be level. It will have a slope of zero. That was the key to this question. The tangent line at the vertex is horizontal --it has a slope of zero. So the point we are looking for has to have d as its y-coordinate.
If you continue in math to calculus, this will be a frequently-used idea: at local extrema, a smooth graph has horizontal tangents. It sounds kind of dry when you say it that way, but it is actually a very useful mathematical fact.
If you would like to see this in more detail, with some animations of a ski riding along a curve, it is something I blogged about a while back here: http://wp.me/p4uvY7-3e
Thanks, I finally got it. 
@pckeller Aw, animation… Can we have Mickey Mouse skiing left to right, right to left, under a dramatic sound track?
Incidentally, Mickey’s velocity vector would be aligned with that tangent line l.
Sorry to disappoint you, @gcf101 --my animation is nothing that dramatic. It just shows a tangent line moving along a graph. But I would not be surprised if someone made something like what you describe. Start with…
https://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land
We just have to add the calculus.
@pckeller Thanks for sharing another math resource! Have to confess, I was not aware of golden ratio built in a pentagon.
I just thought of one more Mickey Mouse (in a good sense) illustration of a line tangent to parabola at its vertex: if whatever the nemesis Mickey has erased half of parabola from the bottom up, Mickey, after skiing all the way down the remaining half of parabola to its vertex, would get airborne and, in the absence of gravity and air resistance, would fly infinitely along that horizontal tangent line.
You don’t need calculus. All you need to know is that a line tangent to a parabola touches it in exactly one point. A line tangent to the vertex of a parabola is horizontal because otherwise it would intersect the parabola in more than one point. (You can see this also by drawing non-horizontal lines through the vertex.) The equation of a horizontal line containing (c,d) is y=d. The only point among the answer choices with y=d is A.
I wasn’t suggesting that you need calculus on the SAT! But this question absolutely makes use of a calculus idea. “Touches in exactly one point” or “touches but does not cross” are helpful intuitive rules of thumb but not really rigorous. Same problem with drawing the tangent line: if you draw it slightly uphill or slightly downhill, it can still look enough like a tangent line to fool someone who did not already know the concept. And someone who already knew the concept probably wouldn’t be asking…
Maybe I did not formulate it well the first time.
Here is the rigorous form:
Symmetry property of the parabola: for every point on the parabola except the vertex, there is exactly one other point with the same y-coordinate. There is no other point on the parabola with the same y-coordinate as the vertex.
From this it follows that a line with the equation y=the y-coordinate of the vertex will intersect the parabola in exactly one point, the vertex. This line cannot intersect the parabola in a second point because there is no other point on the parabola with the y-coordinate of the vertex.
A line that intersects the parabola in exactly one point is tangent to the parabola at that point.
Therefore y=the y-coordinate of the vertex is tangent to the parabola at the vertex.
So all you have to know are the symmetry properties of the parabola and the definition of tangency as intersection in exactly one point. This definition of tangency is from plane geometry, not calculus. I believe both of these concepts have long been in the skill base of the SAT.
Well, if we wanted to be a bit more formal, the line would have to intersect the parabola at (c,d) and the slope of the line must equal f’(c). Otherwise, the vertical line x = c also intersects the parabola at exactly one point, but is not tangent.
But this is a somewhat picky detail, and we don’t need calculus on the SAT.
@MITer94
Ok you got me.
A modification:
The tangent line intersects the parabola in exactly one point and does not pass through any points interior to the parabola.
There has to be a plane geometry definition of a tangent line because tangent lines were around way, way before calculus was invented.
From Wikipedia: “Leibniz defined it as the line through a pair of infinitely close points on the curve.”
However, considering that the problem of finding a tangent line to a curve is one of the problems that led to the development of calculus as we know it today, I’d say the two go hand-in-hand.
Clearly there is more than one historical definition of a tangent line. The problem finding the equation of a tangent line that was solved by calculus was not the problem of finding the tangent to the vertex of a parabola or to a circle – the types of tangent lines we find on the SAT. The latter are plane geometry problems, not calculus problems (although of course you can use calculus for them if you want to). Euclid’s Book of Elements has tangent lines too, but absolutely no calculus.
Just for fun, let me nitpick further…look at the graph of y=x^4-2x^2. Would you say that it has a horizontal tangent line at the origin? I think we agree it does. But that line touches the graph more than once – but not in the neighborhood of the origin. So even the “touch the graph once” definition is not a completely rigorous definition.
There are times when mathematical ideas have developed out of sequence. Calculus is an example. It’s critical concept, the limit, was not rigorously defined until well after Newton and Leibniz. I don’t know enough about the history of geometry to discuss how rigorously Euclid defined tangents to circles.
Again, this is for fun. Not for SAT.
I forgot to say: @Plotinus I agree with you that however we define a tangent line, the symmetry of a parabola does require that the slope at the vertex be zero (or undefined). That’s a nice point.
@Plotinus Taking turns at going nitpicky.
If the question were about a line tangent to the vertex of a half circle that opens upward, that would be a plane plane geometry question, and OP would most likely not have any difficulty surmising that the line had to be horizontal.
Euclude works with tangent lines to circles, not parabolas or other curves, and switching to a parabola propels the original question into a more advanced territory of algebra/precalculus (and, for the mathematically inclined, calculus).
I think we can extend the non-calculus definition of tangent line to local minima/maxima as follows:
A line is tangent to a point that is a local extremum in the interval [a,b] iff
(1) The line intersects the curve in exactly one point in the interval [a,b], i.e, the extremum itself
(2) The line does not contain any points in the “local interior” of the curve (the interior of the curve in the neighborhood of the local extremum).
You are right about Euclid. But the Greeks definitely studied tangents to other conics using plane geometrical methods. For example, Appollonius studied tangencies to parabolas (and not just to the vertex) in his book Conics.
https://en.wikipedia.org/wiki/Tangent
“In Apollonius’ work Conics (ca. 225 BC) he defines a tangent as being a line such that no other straight line could fall between it and the curve.[3]”
See also
http://www.mlahanas.de/Greeks/Conics.htm
“Apollonius used the so-called Symptoms that describes a constant relation between varying magnitudes that depend on the position of an arbitrary point on a curve, example a point C on a parabola. If |AE|= |ED| then AC is a tangent to the parabola”
Hey great nitpicking! It is right to nitpick because the SAT is 90% nitpicking. 
@Ruksha probably looks at this runaway thread and stakes his head in disbelief: dudes, it was just a level 3 math question; where the **** are you going with it?! 
The question from a pedagogical point of view:
How should you explain the solution of this problem to a student?
I would not use the definition of tangency in terms of the derivative. If the student has studied calculus, he or she solved or should have solved the problem yesterday. Also, this is using a bazooka to kill a fly.
If the student has studied pre-calc, he or she might already know that the line tangent to a local extremum is horizontal. But again, I think students who know this should not have trouble with the problem. Or at least, they should not have trouble with determining the slope of a tangent line to the vertex of a parabola.
Instead, I would remind the student of the plane geometry (Euclidean) definition of tangency: The tangent line touches the curve at exactly one point.Then I would ask the student to sketch the parabola and a line that touches the vertex in exactly one point. (If the student draws a vertical line through the vertex as suggested by @MITer94 I would immediately send him home. 
)
If the student draws a horizontal line, I would say “Good, the tangent line is horizontal. Why?” I would try to elicit or otherwise give the answer is that a horizontal line can’t touch the parabola in more than one point because all points on a horizontal line have the same y-value, but there is no other point on the parabola with the same y-value as the vertex. So a horizontal line that contains the vertex cannot contain any other point on the parabola.
If the student is still not sure how to draw the tangent line, I would ask whether a horizontal line through the vertex would touch the parabola in one point or more than one point, and why.
At this point the student knows that a tangent to the vertex of a parabola is horizontal and can use that principle to more easily solve other problems involving the tangent to the vertex. I don’t think it necessary to complicate things any further, or to talk about rates of change, average versus instantaneous rates, limits, etc. However, following @pckeller, we could generalize to the observation that the tangent line to any local maximum or minimum is horizontal.
So yes, this problem is going to be easier for students who have studied calc or pre-calc well, but even sophomores should be able to do it, at least the second time.
Redesigned SAT= excess math verbiage.