91Ó°ÊÓ

Understanding the standard form of a parabola is key to solving many quadratic problems. A parabolic function is typically written as:

f(x) = a(x-h)^{2} + k.

Here, the vertex form of the parabola shows us a lot. The values of h and k are the coordinates of the vertex, while 'a' describes how 'wide' or 'narrow' the parabola is.

- If 'a' is positive, the parabola opens upwards.
- If 'a' is negative, the parabola opens downwards.

Remember, this form allows us to identify the vertex directly and simplifies understanding the parabola's shape and position on the coordinate plane.

The parameters of a parabola play an essential role in its graphical representation. Let’s delve into these parameters:

- **a**: This constant determines the direction and width of the parabola. If |a| is large, the parabola is narrow. If |a| is small, the parabola is wider.
- **h**: This is the x-coordinate of the vertex which shows the horizontal shift from the origin.
- **k**: This is the y-coordinate of the vertex indicating the vertical shift from the origin.

In the function given in the exercise, f(x) = (x + 5)^2 - 8,

we can see that **h = -5** and **k = -8**. Here, 'a' is implicitly 1 (since there is no coefficient in front of the squared term), suggesting a normal parabola that opens upwards.

Determining the vertex of a parabola is straightforward once you know the standard form. Let’s go through the process:

1. Write the given function in the standard form, f(x) = a(x-h)^2 + k.

2. Identify the values of **h** and **k** by comparing both equations.

In our example,
f(x) = (x + 5)^2 - 8,
we can rewrite (x + 5) as (x - (-5)).

3. Hence, **h** = -5 and **k** = -8.
So, the vertex of the parabola is (-5, -8).

Always remember: Identifying **h** and **k** gives you the direct coordinates of the vertex. Plug these values and you've found where the parabola peaks or dips!