Problem 12 - Algebra and Trigonometry

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$ f(x)=-2(x+4)^{2}-8 $$

Step-by-Step Solution

Answer

The vertex of the parabola defined by the function \(f(x) = -2(x+4)^2-8\) is (-4, -8).

Step 1: Identify the vertex form

The given function is written in the vertex form. In the vertex form, \(f(x)=a(x-h)^2+k\), the vertex is at (h, k).

Step 2: Identify the h and k

In our function \(f(x) = -2(x+4)^2-8\), we have \(h=-4\) and \(k=-8\). This means the vertex of the parabola is at (-4, -8).

Key Concepts

Vertex Form of a Quadratic

The vertex form of a quadratic equation provides a clear and direct way to find the vertex of a parabola. It is represented as
\( f(x) = a(x-h)^2 + k \),
where \( (h, k) \) represents the coordinates of the vertex of the parabola, and 'a' determines the width and direction of the parabola. In vertex form, if 'a' is positive, the parabola opens upwards, and if 'a' is negative, as in the given exercise, the parabola opens downwards.

To find the vertex from the vertex form, simply identify the values of 'h' and 'k' in the equation. These numbers are often seen as the opposites of the actual values inside the parentheses with 'x'. For example, in \( f(x) = -2(x+4)^2 - 8 \), 'h' is -4, because \( x+4 \) equates to \(x-(-4)\), and 'k' is -8. Thus, the vertex of the given parabola is at (-4, -8). Understanding this form simplifies finding the vertex to just recognizing the 'h' and 'k' values in the function.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows for the graphical representation of algebraic equations and vice versa, providing a connection between algebra and geometry.

The most common system we use is the Cartesian coordinate system, consisting of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Every point in the plane can be defined by an ordered pair (x, y), representing its coordinates. For parabolas, the vertex is a significant point where the parabola either reaches its maximum or minimum value. In the case of the exercise's equation \( f(x) = -2(x+4)^2 - 8 \), the vertex's coordinates provide crucial information about the parabola's position on the graph as being the point (-4, -8) on the Cartesian plane.

Parabola Characteristics

A parabola is a U-shaped symmetrical curve that can open upwards or downwards, depending on the sign of the coefficient 'a' in its equation. Its most notable characteristics include the vertex, the axis of symmetry, the focus, the directrix, and whether it opens upwards or downwards.

• Vertex: The highest or lowest point on a parabola, dependent on whether it opens up or down. It's the point where the parabola changes direction.
• Axis of Symmetry: A vertical line that runs through the vertex and divides the parabola into two mirror-image halves.
• Focus: A point inside the parabola where all the reflected points converge.
• Directrix: A line perpendicular to the axis of symmetry that helps in defining the curve.

The given quadratic equation, which is in vertex form, makes it easier to determine some of these characteristics, such as the vertex and the axis of symmetry, which in this case is the line \( x = -4 \). Understanding these characteristics helps better analyze and graph the parabola.