The line of symmetry of a vertical parabola with vertex at  has as its equation . In other words, the -coordinate, which is given in Statement 1 but not Statement 2, is the one and only thing needed.

Assume both statements are true, and consider these two equations:

Case 1: 

The -intercept can be proven to be  by substituting 0 for :

We show that the graph intersects the line of equation  twice by substituting 3 for :

The points of intersection are .

To find the -intercept(s), if any exist, substitute 0 for :

This has no real solutions, so the parabola has no -intercepts.

Case 2: 

The -intercept be proved to be  by substituting 0 for :

We show that the graph intersects the line of equation  twice by substituting 3 for :

We examine the discriminant:

The discriminant is positive, so there are two real solutions, meaning that there are two points of intersection. 

To find the -intercept(s), if any exist, substitute 0 for :

We examine this discriminant:

The discriminant is positive, so there are two real solutions, meaning that there are two -intercepts.

Two parabolas have been identified fitting the main condition and those of both statements, but one has no -intercept and one has two. The two statements together provide insufficient information.

The line of symmetry of a vertical parabola is the vertical line passing through the vertex. Each statement alone gives only one point on the graph, neither of which is the vertex, so neither statement alone gives sufficient information.

Now assume both statements to be true. Statement 1 gives one -intercept, ; Statement 2 states that the graph passes through the origin, so it is not only the -intercept, it is also the other -intercept. The -coordinate of the vertex is the arithmetic mean of those of the two -intercepts, so that value is

Only the -coordinate of the vertex is needed to answer the question - we can immediately deduce that the line of symmetry is .

The parabola is concave upward if and only if , and concave downward if and only if . Therefore, we need to know the sign of  to answer the question. We show that the two statements together provide insufficient information by examining two equations.

Case 1: 

; we check the values of the expressions in both statements:

The conditions of both statements are satisfied; since , the parabola that graphs this function is concave upward.

Case 2: 

; we check the values of the expressions in both statements:

The conditions of both statements are satisfied; since , the parabola that graphs this function is concave downward.

The two statements together give the vertex of the parabola, but no clue is given as to the orientation of the parabola - horizontal, vertical, or otherwise. Without that information, or any way to find it, the line of symmetry cannot be determined.

Assume both statements are true.

The parabola is concave upward if and only if , and concave downward if and only if . Therefore, we need to know the sign of  to answer the question. We show that the two statements together provide insufficient information by examining two equations.

Case 1: 

If we substitute 0 for ,  is easily seen to be equal to 6, so the -intercept is . Also:

Case 2: 

If we substitute 0 for ,  is easily seen to be equal to 6, so the -intercept is . Also:

Each equation satisfies the conditions of both statements, but  is positive in one equation and negative in the other.  can assume either sign, so the question of the parabola's concavity is unresolved.

Statement 1): The slope is 4.

Write the slope-intercept form, and substitute the slope.

The point and the y-intercept are unknown.  Either of these will be needed to solve for the graph of this line.

Statement 1) by itself is not sufficient to graph a line.

Statement 2): The y-intercept is 4.

Substitute the y-intercept into the incomplete formula.

The function  can then be graphed on the x-y coordinate plane.

Therefore:

Find the graph of .

I)  is a linear equation which passes through the point .

II)  crosses the y-axis at 1300.

To graph a linear equation, we need some combination of slope, y-intercept, or two points.

Statement I tells us  is linear and gives us one point.

Statement II gives us the y-intercept of .

We can use Statement I and Statement II to find the slope of . Then, we can plot the given points and continue the line in either direction to get our graph.

Slope:

Plugging in the provided value of , 1300, we have the equation of the line :

Find the graph of the linear function .

I)  passes through the points  and .

II)  intercepts the -axis at .

Using I), we can find the slope of the function, and then we can start at either point and extend the slope in either direction to find our graph:

So, using I) we are able to find the slope, from which we can find our graph

II) gives us one point, but without any more information, we cannot use II) by itself to find the rest of the graph

So:

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

You cannot determine the equation of a circle knowing only the center, as in Statement 1. 

But given the coordinates of any diameter of a circle, as in Statement 2, you can use the midpoint formula to find the center, and the distance formula to find the distance from the center to either endpoint - this is the radius. Once you know the center and the radius, just apply them to the standard form of the equation of a circle.