F(x)=x^2+9x-16
what is vertex
axis of semetry

axis of symmetry is .

The ordered pair of the vertex is .

Step-by-step explanation:

Your function is a quadratic.

Compare to .

You should see that .

The x-coordinate of the vertex or the axis of symmetry since the axis symmetry goes through the vertex can be found by computing .

So here we go!

The axis of symmetry is .

When you write your axis of symmetry be sure to write it as an equation.

That is the axis of symmetry is .

Now that was also the x-coordinate of your vertex.  To find the corresponding y-coordinate of the vertex, plug your value for into

.

Put into calculator:

when

The ordered pair of the vertex is .

Vertex:

Axis of symmetry:

Step-by-step explanation:

First I'll find the axis of symmetry. This formula lets us find the a.o.s: .

In , the values of a, b, and c are:

We only need a and b to find the axis of symmetry. Substitute these values into the formula.

Simplify this fraction.

The axis of symmetry of this quadratic function is x = -4.5.

Now to find the vertex, we have to take into account that this quadratic is in standard form, making it a little harder. We have to convert this function into vertex form.

Start by changing f(x) to 'y' and adding 16 to both sides.

Use the completing the square formula:

Keep the balance by adding 20.25 on the left side and adding it on the right side of the equation.

Combine like terms.

Factor the right side of the equation. Ask yourself, "What two numbers multiply to 20.25 (c) and add up to 9 (b)?" These two numbers are 4.5 and 4.5. Rewrite the right side with factors.

Isolate y by subtracting 36.25 from both sides of the equation.

Now this quadratic function is in vertex form, making it super simple to find the vertex using .

Vertex form of a quadratic is:

Compare with the original vertex form and find where h and k are. Those are the x (h) and y (k) values of the vertex.

Since the original vertex form has x - h, the h value in would be a negative since two negatives make a positive. The k value would stay "normal"---negative would mean it is a negative and positive would mean it is a positive number.

Therefore the h value is -4.5, and the k value is -36.25.

The ordered pair of the vertex is .

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

hiking down a winding trail.

step-by-step explanation: