The equation of a parabola can be expressed in the form

,

for some nonzero , where  is the vertex. 

From Statement 1, since  is the only -intercept, it is the vertex. The equation is

,

or, simplified,

for some nonzero . But no further clues are given that could yield the value of .

Statement 2 alone only gives one point, which is not enough to determine the equation of a parabola.

Now assume both statements are true. Then, as stated before, the equation is 

for some nonzero ; we can set up an equation by substituting 0 and 4 for  and , respectively:

The equation is .

Assume both statements are true.

Statement 2 is actually a consequence of Statement 1 - the line of symmetry of a vertical parabola with two -intercepts is the line of the equation , where  is the arithmetic mean of the -coordinates of the -intercepts. Therefore, we only need to examine Statement 1.

Statement 1 alone provides insufficient information, since we can demonstrate that at least two parabolas have the -intercepts given.

Parabola 1:

Substitute 0 for , and solve for :

 or 

and 

Substitute 0 for , and solve for :

 or 

The equations are not equivalent, and both parabolas have -intercepts  and .

Statement 1 alone gives insufficient information; two points alone do not define a parabola. Statement 2 alone  is insufficient; it only establishes that the -intercept is also the vertex.

Assume both statements are true. The equation of a parabola can be expressed in the form

,

for some nonzero , where  is the vertex. Since we know two points that have the same -coordinate, 8, we can take the arithmetic mean of their -coordinates and find the -coordinate of the vertex:

From Statement 2, the parabola has exactly one -intercept, so that -intercept doubles as the vertex, and its -coordinate is 0. 

We now know that the vertex is , and we know that, for some nonzero , the equation of the parabola is

,

or, simplified,

.

We can find  by substituting 8 for both  and :

The equation of the parabola has been determined to be .

Assume Statement 1 alone, which gives three points of the parabola.

The standard form of the equation of a vertical parabola is

for some real numbers , , and , where  is non-zero.

From each of the three given points, the - and -coordinates can be substituted in turn:

or

or 

or

A system of three equations in three variables has been created:

Solving the three-by-three system yields the coefficients of the equation, so Statement 1 alone provides sufficient information.

Statement 2 alone gives only the vertex and the line of symmetry, the latter of which is actually a consequence of the former; however, infinitely many parabolas have their vertices at , so Statement 2 alone provides insufficient information.

The equation of a parabola can be expressed in the form

,

for some nonzero , where  is the vertex. 

From Statement 1 alone,  is the only -intercept; therefore, that is also the vertex. Substituting 4 and 0 for  and , respectively, the equation becomes

,

or, simplified,

.

Since a third point, -intercept  is given, we can substitute 0 and  for  and , respectively, and solve for :

The equation can be determined to be .

Now assume Statement 2 alone. We can examine these two equations:

Case 1: 

We confirm that this parabola passes through  using substitution:

Also, since this can be rewritten as 

,

the vertex is , and the line of symmetry is the line with equation .

Case 2: 

We confirm that this parabola passes through  using substitution:

Also, since this can be rewritten as 

,

the vertex is , and the line of symmetry is the line with equation .

Therefore, Statement 2 alone provides insufficient information.

Given the other endpoint, you can use the distance formula to find the length of the segment:

Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:

The total length is twice that, or 10.

The answer is that either statement alone is sufficient to answer the question.

If you know only that  , then you know that  and , but you still need , or a way finding it.

If you know only that , you still know only that , but you don't know their actual lengths.

If you know both facts, then you know 

Statement I gives us a point.

Statement II gives us the length of the segment. 

We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:

Neither I nor II are sufficient to answer the question. More information is needed.

To find the length of a segment, use the distance formula. The distance formula is given by the following:

Where your 's and 's correspond to the coordinates of the endpoints.

To find the length of Segment YZ, we need the endpoints.

Statement I gives you Point Y's coordinates.

Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.

Therefore, we need both.

Recap:

Find the length of Segment YZ

I) Point Y is located at the point .

II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.

Use Statement II along with Statement I to find the coordinates of Point Z:

Then, use distance formula to find the length of Segment YZ:

To find the length of a segment, we need both endpoints.

Statement I gives us one endpoint.

Statement II relates  and , allowing us to find the second endpoint.

Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.

Using Statement II, we find the second endpoint to be . Use the distance formula to find your answer: