x and y intercept - GMAT Quantitative Reasoning

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 -intercept is the point at which . To find the -coordinate of the -intercept, just substitute 0 for :

Therefore, you need only know ; knowing is neither necessary nor helpful.

If you are given that since . the -intercept is easily determined to be .

The answer is that Statement 2 alone is sufficient, but not Statement 1 alone.

To find the - and y-intercepets, we need both statements of information.

Statement 1 is not enough information because we don't have a reference point for the slope. A slope by itself is not enough to define the line. There are an infinite amount of lines with a slope of 0.6.

Statement 2 is not enough information by itself since it only tells us about 1 point on the line. Again, there are an infinite number of lines that pass through the point (10,2).

Only by using both statements can we find the - and y-intercepts. Solving, we see the line is actually

The -intercept of the graph of an equation is the point at which , so we evaluate :

The -intercept is simply the point , so knowing is necessary, and knowing is neither necessary nor helpful.

To find the -intercept of , evaluate :

Knowing is necessary and sufficient; the value of is irrelevant.

To find the -intercept of , evaluate :

Knowing both and is necessary and sufficient.

Two points are necessary and sufficient to define a line. Therefore, neither statement alone is sufficient to determine the line, but both are sufficient. Once the line is defined, the -intercept - the point at which the line intersects the -axis - can be determined.

The -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is . Statement 1 does not give us this value, but Statement 2 does.

The two statements together prvide insufficient information.

Assume both statements are true. By Statement 1, is a constantly increasing function, so it can intersect the -axis at most one time.

Now examine these two cases.

Case 1:

.

Also, if , then , so .

Since , the function has exactly one -intercept.

Case 2:

Also, if , then , so .

However, 2 raised to any power must be positive, so there is no value for which . The function has no -intercepts.

Statement 1 alone establishes that is always increasing. Its graph cannot have more than one -intercept; if it does, then the graph of the function must have a vertex between two intercepts, violating this statement. But it does not answer the question as to how many intercepts has, as seen in these two cases:

Case 1:

This is a linear function that is always increasing—it is in slope-intercept form, and its slope is 1, a positive number. The graph of has exactly one -intercept.

Case 2:

An exponential function with a base greater than 1, such as this, is an increasing function; however, 2 raised to any power must be positive, so there is no value for which . The graph of has no -intercepts.

Statement 2 alone establishes that at least one -intercept exists - since , is an -intercept. It does not, however, rule out the possibility of more -intercepts.

Now assume both statements are true. Since Statement 1 establishes that there is at most one -intercept, and Statement establishes that there is at least one -intercept, the two statements together establish that there is exactly one.

Assume both statements are true.

By Statement 1, is decreasing on the domain interval ; by Statement 2, is increasing on the domain interval . Therefore, must have its minimum value when .

This does not, however, tell us the number of -intercepts. For example, the graph of has as its minimum point , and, subsequently, exactly one -intercept. The graph of has as its minimum point and, subsequently, no -intercepts.

The -intercept of the graph of is the point at which it intersects the -axis. Since this point has -coordinate 0, the -coordinate is the value for which . Statement 1 gives us this value; Statement 2 does not.

Statement 1 provides insufficient information, since the slope of the line alone is not enough from which to deduce the -intercept. Statement 2 alone tells us that the line passes through the point ; since this is on the -axis, this is the -intercept.