Let  stand for the values Stanley wrote in the square, the circle, and the triangle, respectively. The equation becomes

.

To find the -coordinate of the -intercept, set  and solve for :

It is therefore necessary and sufficient to know the values of  and  - the values Ralph wrote in the square and the triangle - in order to determine the -intercept of Ralph's equation. Each statement alone give only one of those numbers; the two together give both.

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 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.

If we let  stand for the number Ray wrote in the square and  stand for the number he wrote in the circle, the equation becomes

,

the slope-intercept form of the equation of a line. In this form, the -intercept is solely determined by the value of —namely, the value that Ray wrote in the circle. Statement 1 is therefore irrelevant, and Statement 2 alone establishes that Ray did not succeed.

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.

Let  stand for the values Ralph wrote in the square, the circle, and the triangle, respectively. The equation becomes

.

To find the -coordinate of the -intercept, set  and solve for :

It is therefore necessary and sufficient to know the values of  and  - the values Ralph wrote in the circle and the triangle - in order to determine the -intercept of Ralph's equation. Statement 1 is irrelevant, and Statement 2 only provides the value Ralph wrote in the triangle.

If we let  stand for the number Gloria wrote in the square and  stand for the number she wrote in the circle, the equation becomes

,

the slope-intercept form of the equation of a line. We can find the -coordinate of the -intercept by setting  and solving for :

Therefore, it is necessary and sufficient to know both  and  - both of the numbers Gloria wrote - to determine the -intercept of the equation she made. Neither statement provides both values, but both statements together do.

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.