Parabolas are common mathematical objects that are often encountered in different disciplines including physics, engineering, and architecture. One of the most defining features of a is its , which is a point where the curve changes direction. The vertex is also important for determining other characteristics of a parabola such as its axis of symmetry, its minimum or maximum value, and its roots. In this article, we will discuss how to find the vertex of a parabola using different methods.

The standard form of a parabola is:

y = ax^2 + bx + c

where a, b, and c are constants that determine the shape and location of the parabola. To find the vertex of a parabola in standard form, you can use the formula:

x = -b/2a

y = f(x)

where f(x) is the value of y when x is substituted into the equation of the parabola. The first formula gives you the x-coordinate of the vertex while the second formula gives you the corresponding y-coordinate.

Let’s illustrate this method using an example. Suppose we have the equation:

y = 2x^2 + 4x – 6

To find the vertex, we need to first identify the values of a, b, and c. In this case, a = 2, b = 4, and c = -6. Substituting these values into the vertex formula, we get:

x = -4/(2*2) = -1

To find the corresponding y-coordinate, we just need to substitute x = -1 into the equation of the parabola:

y = 2(-1)^2 + 4(-1) – 6 = -8

Therefore, the vertex of the parabola is (-1, -8).

Another way to find the vertex of a parabola is to complete the square. This method involves rewriting the equation of the parabola in a form that allows us to directly read the vertex coordinates. The steps for completing the square are as follows:

1. Move the constant term to the right side of the equation:

y – c = ax^2 + bx

2. Factor out the value of a from the terms on the right side:

y – c = a(x^2 + (b/a)x)

3. Add and subtract the square of half the coefficient of the x-term, which is (b/2a)^2:

y – c + (b/2a)^2 = a(x^2 + (b/a)x + (b/2a)^2 – (b/2a)^2)

4. Factor the expression inside the parentheses as a perfect square trinomial:

y – c + (b/2a)^2 = a(x + b/2a)^2 – a(b/2a)^2

5. Simplify the right side and combine like terms:

y – c + (b/2a)^2 = a(x + b/2a)^2 – b^2/4a

Now we have the equation of the parabola in vertex form:

y – c = a(x – h)^2 + k

where h = -b/2a and k = c – b^2/4a are the coordinates of the vertex. Note that the value of a determines the direction of the parabola (upward or downward), and the distance between the vertex and the focus of the parabola.

Let’s use the same example as before to apply the completing the square method:

y = 2x^2 + 4x – 6

Move the constant term to the right side:

y + 6 = 2x^2 + 4x

Factor out the value of a:

y + 6 = 2(x^2 + 2x)

Add and subtract (2/2)^2 = 1 to the right side:

y + 6 + 1 – 1 = 2(x^2 + 2x + 1) – 2(1)

Factor the perfect square trinomial:

y + 7 = 2(x + 1)^2 – 2

Simplify and rewrite in vertex form:

y = 2(x + 1)^2 – 9

Therefore, the vertex of the parabola is (-1, -9).

In conclusion, there are different methods for the vertex of a parabola, depending on the form of the equation and the tools at your disposal. The vertex is a crucial point on the parabola that helps us understand its behavior and characteristics. By using the formulas for the x and y-coordinates, or by completing the square to obtain the vertex form, we can locate the vertex with precision and ease.

Quest'articolo è stato scritto a titolo esclusivamente informativo e di divulgazione. Per esso non è possibile garantire che sia esente da errori o inesattezze, per cui l’amministratore di questo Sito non assume alcuna responsabilità come indicato nelle note legali pubblicate in Termini e Condizioni
Quanto è stato utile questo articolo?
0Vota per primo questo articolo!