Explanation:

The slope of lineis

From statement (1) we get another function of和. Therefore, we can calculate the values of和.

From

we can get.

Pluginto, then we can get

和

Statement (2) only tells us the value of, which is useless to get the value of, because we have three unknown numbers with only two equations given.

Explanation:

每个语句parabo单独给一个点la, which, even it is known to be the vertex, is not enough to determine its equation.

Assume both statements are true. The equation of a vertical parabola with vertexcan be written as

Statement 1 gives the values of和as 2 and 4, respectively, so the equation of the parabola is

for some.

From Statement 2, we can substitute 4 and 7 for和, respectively, and solve

or

for, yielding the complete equation:

This makes the equation.

Explanation:

Statement 1 alone only gives one point of the parabola, which by itself does not determine its equation.

Statement 2 alone only gives two-intercepts, which are not sufficient to determine its equation; for example, the equations

和

are both equations of parabolas with their-intercepts at和.

Assume both statements are true. The standard form of the equation of a vertical parabola is

for some real, whereis nonzero.

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 the two statements together provide sufficient information.

Explanation:

The equation of a parabola can be expressed in the form

,

for some nonzero, whereis the vertex.

From Statement 1, sinceis 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和, respectively:

The equation is.

Explanation:

Assume both statements are true.

声明2实际上是一个consequence of Statement 1 - the line of symmetry of a vertical parabola with two-intercepts is the line of the equation, whereis 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

和

Substitute 0 for, and solve for:

or

方程并不等价,parabolas have-intercepts和.

Explanation:

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, whereis 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 findby substituting 8 for both和:

The equation of the parabola has been determined to be.

Explanation:

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, whereis 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.

Explanation:

The equation of a parabola can be expressed in the form

,

for some nonzero, whereis the vertex.

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

,

or, simplified,

.

Since a third point,-interceptis given, we can substitute 0 andfor和, 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 throughusing 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 throughusing 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.