Problem 12 - College Algebra

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 coordinates of the vertex of the parabola are (-4, -8).

Step 1: Identify vertex from vertex form

\(f(x) = -2(x+4)^2 - 8\) is in vertex form \(f(x) = a(x-h)^2 + k\) with \(h = -4\) and \(k = -8\).

Step 2: State the vertex

The vertex is \((-4, -8)\). Since \(a = -2 < 0\), the parabola opens downward.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is characterized by the highest degree term being squared. It has the standard form: \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The parabolic shape created by this function makes it unique among other polynomials.
Quadratic functions are used frequently in various mathematical models because they depict relationships that change at a constant rate. Their graphs show how one variable affects another and are fundamental in physics, engineering, and economics to name a few.
Understanding how to work with these functions is vital for solving real-world problems that involve trends and predictions.

Parabola

A parabola is the U-shaped curve one observes when graphing a quadratic function. It is symmetrical, and the vertex of this shape provides crucial information about the function’s maximum or minimum value.
Every parabola has an axis of symmetry, a vertical line that divides it into two mirror-image halves. This line always passes through the vertex of the parabola. Depending on the coefficient \(a\) in the quadratic function, the parabola can either open upwards if \(a > 0\), or downwards if \(a < 0\).
Parabolas are not just limited to mathematics; they can be seen in various physical structures, such as satellite dishes and bridges, where the parabolic shape is used to direct or spread forces.

Vertex Form

The vertex form of a quadratic function is a way to express the equation that clearly shows the vertex of the parabola. It takes the form \( f(x) = a(x-h)^2 + k \), where \((h, k)\) are the coordinates of the vertex. This form is particularly useful when you want to easily identify the vertex and understand how the parabola is positioned on a graph.
Changing the values of \(h\) and \(k\) shifts the parabola horizontally and vertically on the coordinate plane, respectively. The value of \(a\) affects the width and the direction in which the parabola opens.

• If \(|a| > 1\), the parabola becomes narrower.
• If \(0 < |a| < 1\), it becomes wider.

Understanding the vertex form is key for anyone looking to graph quadratic functions or analyze their properties.

Coordinates of the Vertex

The coordinates of the vertex are essential in understanding the graph of a quadratic function, as they represent the highest or lowest point of the parabola. In vertex form, \( f(x) = a(x-h)^2+k \), the vertex is simply the point \((h, k)\).
In the given quadratic function \( f(x) = -2(x+4)^2 - 8 \), we identified the vertex as \((-4, -8)\). Here, \(h = -4\) and \(k = -8\). This means the vertex of the parabola is located at \((-4, -8)\) on the coordinate plane.
Identifying these coordinates helps in quickly sketching the graph and understanding the behavior of the function, such as where it reaches its minimum or maximum value. This point is crucial for solving optimization problems and other mathematical applications.