DSQ: Calculating the equation of a curve - GMAT Quantitative

The slope of line is

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

From

we can get .

Plug into , then we can get

and

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.

Each statement alone gives one point of the parabola, 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 vertex can be written as

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

for some .

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

or

for , yielding the complete equation:

This makes the equation .

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

and

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

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

for some real , where is 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.

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 .

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.

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 .

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 .

The slope of line is

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

From

we can get .

Plug into , then we can get

and

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.

Each statement alone gives one point of the parabola, 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 vertex can be written as

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

for some .

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

or

for , yielding the complete equation:

This makes the equation .

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

and

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

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

for some real , where is 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.

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 .

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.

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 .

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 .

The slope of line is

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

From

we can get .

Plug into , then we can get

and

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.

Each statement alone gives one point of the parabola, 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 vertex can be written as

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

for some .

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

or

for , yielding the complete equation:

This makes the equation .

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

and

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

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

for some real , where is 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.

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 .