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.

In order to be in standard form, the equation for a parabola must be written in one of the following ways:

We can see that our equation has all of the components of the first form above, so now all we must do is some algebra to rearrange the equation and express the function y in terms of x. We start by simplifying the fraction on the left side of the equation, and then we isolate y to give us the equation of the parabola in standard form:

The standard form of a parabola is .

Factorize the right side of , and simplify.

First, write the equation of the parabola in standard form.

Determine the values of the coefficients.  The value of the y-intercept is 4, which means that .

Write the vertex formula.

The given point of the vertex is , which indicates that:

Substitute the value of the point and  into the standard form.

Substitute this value into  to determine the value of .

Substitute the values of  coefficients into the standard form of the parabola.

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

Factor both sides of the equation to get the standard form of a horizontal parabola.

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

Factor both sides of the equation to get the standard form of a horizontal parabola.

Recall the standard equation of a horizontal parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

Factor both sides of the equation to get the standard form of a horizontal parabola.

Recall the standard equation of a vertical parabola:

, where  is the vertex and  is the focal length.

Start by isolating the  terms.

Complete the square on the left. Make sure to add the same amount to both sides of the equation!

Factor both sides of the equation to get the standard form of a vertical parabola.