<p>I saw this on Collegeboard. The</a> Official SAT Question of the Day
Here is the explanation:
The fact that (x) less than or equal to u for all values of x means that the point P(t, u) is the maximum of the quadratic function , so P(t, u) must be the vertex of the parabola. The parabola intersects the x-axis at the points (1, 0) and (4, 0), so t must be halfway between 1 and 4. Therefore, t = 1 + (4 minus 1) over 2 = 2.5. Of the choices, the only one with x-coordinate equal to 2.5 is (2.5, 3). Therefore, (2.5, 3) could be the coordinates of point P.</p>
<p>This is the part I don't understand:
he parabola intersects the x-axis at the points (1, 0) and (4, 0), so t must be halfway between 1 and 4. Therefore, t = 1 + (4 minus 1) over 2 = 2.5.</p>
<p>CB's explanation pretty much covers it, assuming you know basic traits of parabolas. Because f(x) is less than or equal than u for all values x, point P must be the vertex of the parabola. All parabolas are symmetrical down the line the line of symmetry which always passes through the vertex. Therefore, the 2 zero's must be same distance from the vertex. Therefore, the X coordinate of the vertex is half way between the zeroes.</p>
<p>the parabola is a function that grows or declines, but ones it reaches its zenith it goes the oppostite way, so the one side of a parabola has to be the inverse of the other, therefore has to be the same distance.</p>
<p>I saw this question the other day, didn't answer it, and forgot to go back to do it again. My method of finding the x-coordinate involved working backwards. I transformed x=1 and x=4 into x-1=0 and x-4=0. From there, I multipled to form x²-5x-4 and used the axis of symmetry formula, x=-b/2a to find what I needed to know.</p>
<p>
[quote]
Hi,
Why is that t = 1 + (4 minus 1) over 2 = 2.5 (CB explanation) Why shoudn't it be (4+1)/2=2.5 ?
[/quote]
Either way works. CB doesn't always have the best method, and quite frankly, I think your method is easier to understand and makes more sense.</p>
<p>CB says its way because one zero is at 1,0 and the other is at 4,0. Thus, the midpoint between 1 and 4 is the distance between 1 and 4 divided by 2 then plus 1 to shift it over. If you don't get what I mean, don't worry about it because CB's method of finding the middle between the 2 points is stupid in convoluted. Just take the average of the 2 numbers like you did.</p>
<p>There are no tic marks on the y-axis.
An assumption that the x-axis and y-axis have the same scale does not follow from the fact that there is no note "Not drawn to scale" .</p>
<p>Therefore, f(x) does not have to be -(x-1)(x-4):
f(x) = a(x-1)(x-4) where a<0; |a| can be as big as you want, making all five answer choices correct, since all five given x coordinates are between 1 and 4.</p>
<p>"CB says its way because one zero is at 1,0 and the other is at 4,0. Thus, the midpoint between 1 and 4 is the distance between 1 and 4 divided by 2 then plus 1 to shift it over. If you don't get what I mean, don't worry about it because CB's method of finding the middle between the 2 points is stupid in convoluted. Just take the average of the 2 numbers like you did."</p>
<p>I don't get it still. I know, I'm just being curious. Why should we shift 1? Is that some formula we never heard of?</p>
<p>Because we are doing this the hard way. There are two different ways to do this. Both are correct. One is easier.</p>
<p>Either way, you are looking for the x-coordinate of a point midway between the two given x-intercepts. Those two x-coordinates are 1 and 4.</p>
<p>EASY WAY: Find the midpoint by averaging 1 and 4: That's (4 + 1) /2</p>
<p>LONGER WAY: Find the distance between the two points: That's 4 - 1 = 3.
Then divide that distance in half. So now you have the distance from either point to the midpoint. Then, finally, add that distance to the left endpoint. You get the same answer as the easy way. Of course, if your goal is to make things more complicated, you could SUBTRACT that distance from the right endpoint. That works too. I recommend the easy way.</p>