Problem 10 - College Algebra

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)--3(x-2)^{2}+12$$

Step-by-Step Solution

Answer

The vertex of the parabola defined by the function \(f(x) = -3(x - 2)^2 + 12\) is at the point (2, 12).

Step 1: Identify a, h, and k from the Function

The given quadratic function is \(f(x) = -3(x - 2)^2 + 12\). Compared to the standard form of a parabola, \(f(x) = a (x - h)^2 + k\), we can identify that \(a = -3\), \(h = 2\), and \(k = 12\)

Step 2: Calculate the Vertex

The vertex of a quadratic function is given by the point \((h, k)\). From step 1, we have already identified that \(h = 2\) and \(k = 12\). Therefore, the vertex of the given function is at the point \((2, 12)\).

Key Concepts

Quadratic Function

A quadratic function represents one of the simplest forms of polynomial functions. It is typically expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the world of mathematics, these functions graph as a curve called a parabola. Parabolas display a symmetrical shape which can open upwards or downwards, depending on the sign of \( a \).

• If \( a \) is positive, the parabola opens upwards, resembling a "U" shape.
• If \( a \) is negative, the parabola opens downwards, resembling an inverted "U".

The quadratic function is key in many real-life applications, from physics to engineering, as it can describe projectile motions, structures, and many paths of objects.

Standard Form of a Parabola

The standard form of a parabola simplifies the process of identifying important features such as the vertex. It's expressed as \( f(x) = a(x - h)^2 + k \), where \( a \), \( h \), and \( k \) have specific meanings.

• The parameter \( a \) still determines the direction of the parabola. A positive \( a \) indicates it opens upwards, while a negative \( a \) shows it opens downwards.
• Parameters \( h \) and \( k \) are especially important because they represent the vertex's coordinates, \( (h, k) \).

In the given problem, the function \( f(x) = -3(x - 2)^2 + 12 \) is already in this clean format, making it easy to identify \( a \), \( h \), and \( k \). This form eliminates complex calculations, allowing for almost instant reading of the vertex and parabola 'behavior.'

Coordinates of the Vertex

Finding the coordinates of the vertex is a straightforward yet crucial step in analyzing quadratic functions of parabolas. The vertex provides the highest or lowest point on the parabola, significant for understanding the function's range.

To find these coordinates, one need only compare the quadratic function to its standard form \( f(x) = a(x - h)^2 + k \). In the problem at hand, this process helps us identify:

• \( h = 2 \)
• \( k = 12 \)

Thus, the vertex is \((2, 12)\). The implications of these coordinates dictate the maximum or minimum values of the function and provide a central point in its symmetry. This central figure is key to graphing and making predictive analyses in algebra and calculus.