Let's start by looking at the given equation:

The inequality is written in slope-intercept form; therefore, the slope is equal to  and the y-intercept is equal to .

All of the graphs depict a line with slope of  and y-intercept . Next, we need to decide if we should shade above or below the line. To do this, we can determine if the statement is true using the origin . If the origin satisfies the inequality, we will know to shade below the line. Substitute the values into the given equation and solve.

Because this statement is true, the origin must be included in the shaded region, so we shade below the line.

Finally, a statement that is "less than" or "greater than" requires a dashed line in the graph. On the other hand, those that are "greater than or equal to" or "less than or equal to" require a solid line. We will select the graph with shading below a dashed line.

The -coordinate of the vertex of a parabola of the form 

is

.

Set :

The -coordinate is therefore :

, which is the correct choice.

Set  and solve for :

The terms have a GCF of 2, so

The trinomial in parentheses can be FOILed out by noting that  and :

Set each of the linear binomials to 0 and solve for :

or

The parabola has as its two intercepts the points  and .

A parabola that opens downward has the general formula

,

as the negative sign in front of the  term makes flips the parabola about the horizontal axis.  

By contrast, a parabola of the form rotates about the vertical axis, not the horizontal axis. 

Therefore, is not the equation for a parabola that opens downward.

For any parabola, the general equation is

, and the x-coordinate of its vertex is given by

.

For the given problem, the x-coordinate is

.

To find the y-coordinate, plug  into the original equation:

Therefore the vertex is at .

Parabolas can either be in the form

for vertical parabolas or in the form

for horizontal parabolas. Since the equation that the problem gives us has a y-squared term, but not an x-squared term, we know this is a horizontal parabola. The rules for a horizontal parabola are as follows:

• If , then the horizontal parabola opens to the right.
• If , then the horizontal parabola opens to the left.

In this case, the coefficient in front of the y-squared term is going to be positive, once we isolate x. That makes this a horizontal parabola that opens to the right.

Factor 2 as GCF from the first two terms giving us:

Now we complete the square by adding 4 to the expression inside the parenthesis and subtracting 8 ( because ) resulting in the following equation:

which is equal to

Hence the vertex is located at

A parabola is one example of a quadratic function, regardless of whether it points upwards or downwards.

The red line represents a quadratic function and will have a formula similar to .

The blue line represents a linear function and will have a formula similar to .

The green line represents an exponential function and will have a formula similar to .

The purple line represents an absolute value function and will have a formula similar to .

We can determine if a parabola is upward or downward facing by looking at the coefficient of the  term. It will be downward facing if and only if this coefficient is negative. Be careful about the answer choice . Recall that this means that the entire value inside the parentheses will be squared. And, a negative times a negative yields a positive. Thus, this is equivalent to . Therefore, our answer has to be . 

The equation of a parabola can be written in vertex form: .

The point  in this format is the vertex. If  is a postive number the vertex is a minimum, and if  is a negative number the vertex is a maximum.

In this example, . The positive value means the vertex is a minimum.