The equation for the eccentricity of an ellipse is , where  is eccentricity,  is the distance from the foci to the center, and  is the square root of the larger of our two denominators.

Our denominators are  and , so .

To find , we must use the equation , where  is the square root of the smaller of our two denominators.

This gives us , so .

Therefore, we can see that

 .

In order to determine which direction the parabola opens, we must first put the equation in standard form, which can be expressed in one of the following two ways:

If the equation is for  as in the first above, the parabola opens up if  is positive and down if is negative. If the equation is for as in the second above, the parabola opens right if is positive and left if is negative. Rearranging our equation, we get:

We can see that our equation is for , which means the parabola will open either left or right. The sign of the first term is negative, so this parabola will open to the left.

For the function

The parabola opens upwards if a>0

and downards for a<0

Because 

The parabola opens upwards.

The standard form for a parabola is in the form: 

The coefficient of the  term determines whether if the parabola opens upward or downward.  Since the  term in the function  is , the parabola will open downward.

Use the FOIL method to determine  in its standard form for parabolas, which is .  

Regroup the terms.

Since the coefficient of the  term is negative, the parabola will open downward.

The focus is above the vertex, which means that the parabola will open up

In order to determine which way this parabola, group the variables in one side of the equation.  Add  on both sides of the equation to isolate .

Because the equation is in terms of , the parabola will either open left or right.  Notice that the coefficient of the term is negative.

The parabola will open to the left.

Determine whether the following parabola opens up or down and state how you know.

To determine the direction a parabola opens, we only need to worry about the squared term. 

In this case, it is positive, so the parabola opens upward.

The linear term is negative, so the parabola will be to the right of the y-axis.

The constant term is negative, so the parabola will be located below the x-axis.

In order to determine how the parabola will open, we will need to rewrite the equation in standard form.  

Write the standard form for parabolas.

Subtract  from both sides of the equation.

Since the coefficient of the  is negative, and the equation is in terms of , the parabola will open downward.

The answer is:  

The answer is because it is the only degree-2 polynomial with a negative leading coefficient.