Quadratic Functions - Algebra II

Start by visualizing the graph associated with the function :

Terms within the parentheses associated with the squared x-variable will shift the parabola horizontally, while terms outside of the parentheses will shift the parabola vertically. In the provided equation, 2 is located outside of the parentheses and is subtracted from the terms located within the parentheses; therefore, the parabola in the graph will shift down by 2 units. A simplified graph of looks like this:

Remember that there is also a term within the parentheses. Within the parentheses, 1 is subtracted from the x-variable; thus, the parabola in the graph will shift to the right by 1 unit. As a result, the following graph matches the given function :

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

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.

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 .

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.

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 graph of a quadratic function has an -intercept at any point at which , so, first, set the quadratic expression equal to 0:

The number of -intercepts of the graph is equal to the number of real zeroes of the above equation, which can be determined by evaluating the discriminant of the equation, . Set , and evaluate:

The discriminant is negative, so the equation has two solutions, neither of which are real. Consequently, the graph of the function has no -intercepts.

The vertex form of a quadratic equation is given by

Where the vertex is located at

giving us .

We can determine the lowest possible value of the expression by finding the -coordinate of the vertex of the parabola graphed from the equation . This is done by rewriting the equation in vertex form.

The vertex of the parabola is the point .

The parabola is concave upward (its quadratic coefficient is positive), so represents the minimum value of . This is our answer.

A parabola is a curve that can be represented by a quadratic equation. The only quadratic here is represented by the function , while the others represent straight lines, circles, and other curves.

From the vertex, we know that the equation of the parabola will take the form for some .

To calculate that , we plug in the values from the other point we are given, , and solve for :

Now the equation is . This is not an answer choice, so we need to rewrite it in some way.

Expand the squared term:

Distribute the fraction through the parentheses:

Combine like terms:

This is a quadratic function. The -coordinate of the vertex of the parabola can be determined using the formula , setting :

Now evaluate the function at :

To find the x-intercepts of the equation, we set the numerator equal to zero.

Vertex of quadratic equation is given by .

For ,

,

so the coordinate of vertex is .

Assume y=0,

,

In order to find the vertex of a parabola, our first step is to find the x-coordinate of its center. If the equation of a parabola has the following form:

Then the x-coordinate of its center is given by the following formula:

For the parabola described in the problem, a=-2 and b=-12, so our center is at:

Now that we know the x-coordinate of the parabola's center, we can simply plug this value into the function to find the y-coordinate of the vertex:

So the vertex of the parabola given in the problem is at the point

To find the max or min of , use the vertex formula and substitute the appropriate coefficients.

Since the leading coefficient of is negative, the parabola opens down, which indicates that there will be a maximum.

The answer is:

To simplify , determine the factors of the first and last term.

The factor possibilities of :

The factor possibilities of :

Determine the signs. Since there is a positive ending term and a negative middle term, the signs of the binomials must be both negative. Write the pair of parenthesis.

These factors must be manipulated by trial and error to determine the middle term.

The correct selection is .

The answer is .