f(x)=-x^2 - 3x

Find the axis of symmetry, vertex, direction it opens, and write the equation in vertex form.

If f(x) =x^2 + 2x
g(x)= x-9
h(x) = 3x - 2
Find [f o g o h] (-2)

Please explain how you got it I'm really confused right now.

You have to know how to complete the square.

f(x) = -x^2 - 3x
= -(x^2 + 3x +9/4 - 9/4)
= -( (x + 3/2)^2 - 9/4)
= -(x + 3/2)^2 + 9/4

take it from there, all the information is just there in front of you

for you 2nd problem, [f o g o h] (-2) means
f(g(h(-2))) to me
= f(g(-8))
= f(-17)
= (-17)^2 + 2(-17)
= 255

To find the axis of symmetry, vertex, and direction of opening of a quadratic function, we can take the given quadratic function f(x) = -x^2 - 3x and first determine its vertex form.

1. Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.

2. Determining the Vertex:
To find the vertex of the given function, we need to convert f(x) = -x^2 - 3x into vertex form. We can do this by completing the square.

f(x) = -x^2 - 3x

First, factor out the common negative sign: f(x) = -(x^2 + 3x)

Next, complete the square inside the parentheses.

To complete the square, take half of the coefficient of x (which is 3/2) and square it: (3/2)^2 = 9/4.

Add this value inside the parentheses, while also subtracting the same value multiplied by the leading coefficient (which is -1).

f(x) = -(x^2 + 3x + 9/4 - 9/4)

Simplify: f(x) = -(x^2 + 3x + 9/4) + 9/4

Factor the perfect square trinomial: f(x) = -(x + 3/2)^2 + 9/4

So the vertex form of the function is f(x) = -(x + 3/2)^2 + 9/4.

3. Axis of Symmetry and Vertex:
From the vertex form, we can determine the vertex as (-3/2, 9/4). The axis of symmetry is the vertical line passing through the vertex, which is x = -3/2.

4. Direction of Opening:
Since the coefficient of x^2 is negative (-1 in this case), the parabola opens downward.

Thus, for the given quadratic function f(x) = -x^2 - 3x:
- Axis of symmetry: x = -3/2
- Vertex: (-3/2, 9/4)
- Direction of opening: Downward

Now, let's move on to the second part of your question:

If f(x) = x^2 + 2x,
g(x) = x - 9,
h(x) = 3x - 2,

we need to find [f o g o h] (-2), which represents the composition of these three functions evaluated at x = -2.

To find [f o g o h] (-2), we perform the composition in the following order: h(x), then g(x), and finally f(x).

1. Evaluate h(x) = 3x - 2 at x = -2:
h(-2) = 3(-2) - 2
= -6 - 2
= -8

2. Evaluate g(x) = x - 9 at x = -8 (the result obtained from h(x)):
g(-8) = -8 - 9
= -17

3. Evaluate f(x) = x^2 + 2x at x = -17 (the result obtained from g(x)):
f(-17) = (-17)^2 + 2(-17)
= 289 - 34
= 255

Therefore, [f o g o h] (-2) = 255.

By following these steps, you should be able to find the solution and understand the process. Let me know if you have any further questions!