Problem 9 - College Algebra

Question

In Exercises \(9-16,\) find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)-2(x-3)^{2}+1$$

Step-by-Step Solution

Answer

The coordinates of the vertex of the given parabola are (3, 1).

Step 1: Identify the form of the given equation

The given quadratic equation is already in vertex form which is indicated as \(f(x) = a(x-h)^{2}+k\).

Step 2: Identify the vertex of the parabola

In the vertex form equation, \((h,k)\) are the coordinates of the vertex. Comparing this with the given equation \(f(x) = 2(x-3)^{2}+1\), we can identify \(h=3\) and \(k=1\).

Key Concepts

Parabola Vertex

The vertex of a parabola is a crucial point in its structure. It represents either the highest or lowest point of the parabola, depending on its orientation. For parabolas that open upwards, the vertex is the minimum point, while for those that open downwards, it's the maximum. Understanding the vertex is central to grasping how a parabola behaves on a graph.
The vertex is particularly important in applications such as physics and economics, where it can represent equilibrium points or extremes in functions. In graphing, it can help us quickly sketch a parabola by providing the central turning point.

Quadratic Function

A quadratic function is a polynomial function of degree two, generally expressed as \( f(x) = ax^2 + bx + c \). However, it can also be represented in vertex form, \( f(x) = a(x-h)^2 + k \). Here, "\(a\)" indicates the parabola's direction and width, while \((h, k)\) is the vertex.
Quadratic functions model numerous real-world phenomena, from projectile motion in physics to profit maximization in economics. Their graphs are parabolas, and understanding their structure helps in analyzing situations described by them. With vertex form, you can define the vertex directly, making it easy to identify and graph the parabola.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to study points, lines, and curves on a graph. It provides a systematic approach to discussing geometric elements and their relationships through an algebraic lens.
With respect to quadratic functions and parabolas, coordinate geometry helps us understand how these graphs interact with the axes and other geometric constructs. Calculating slopes of tangents, finding points of intersection, and determining the distances between points on a graph all involve coordinates. By using equations and formulas, we can make the concepts of geometry more concrete and solution-oriented.

Vertex Coordinates

Vertex coordinates are simply the coordinates \((h, k)\) from the vertex form of a quadratic equation \(f(x) = a(x-h)^2+k\). Identifying these coordinates is straightforward when the equation is in vertex form.

• \(h\): Represents the x-coordinate of the vertex. It indicates the horizontal shift from the origin on the graph.
• \(k\): Represents the y-coordinate of the vertex. It indicates the vertical shift on the graph.

For example, in the equation \(f(x) = 2(x-3)^2+1\), the vertex is at \((3, 1)\). Understanding vertex coordinates allows you to quickly identify and plot the central point of a parabola on a graph, facilitating further analysis.