Problem 9 - Algebra and Trigonometry

Question

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 vertex of the parabola defined by the function \( f(x)=2(x-3)^{2}+1 \) is at point (3,1).

Step 1: Identify the form of the equation

The given equation is already in the form \( f(x)=a(x-h)^{2}+k \). This is otherwise known as the vertex form of a quadratic equation, as it allows us to directly identify the vertex of the parabola.

Step 2: Recognize the coordinates of the vertex

In our equation \( f(x)=2(x-3)^{2}+1 \), the value of 'h' is 3 and the value of 'k' is 1. Therefore, the vertex of our parabola is located at point (h,k) or in this particular case, at point (3,1).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree is 2, which is why it's also called a second-degree polynomial. It's typically written in the standard form:

• \( f(x) = ax^2 + bx + c \)
Such functions graph as a parabola on the Cartesian plane. These curves can open upwards or downwards, depending on the sign of the coefficient \(a\):
• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Quadratic functions are essential for many real-life applications such as physics, engineering, and economics, as they often describe natural phenomena and various optimization problems.

Vertex Form

The vertex form of a quadratic function simplifies the process of identifying the vertex of a parabola. The vertex form is expressed as:

• \( f(x) = a(x-h)^2 + k \)

This format is useful because:

• The coordinates \((h, k)\) give the vertex of the parabola directly.
• It shows the translation of the graph from the parent function \(y = ax^2\).

In comparing the vertex form to the standard form \(y = ax^2 + bx + c\), you'll often convert between the two to find different properties or to perform various calculations.

Coordinates

Coordinates are essential in graphing, providing exact locations for points on a plane. Each point is expressed as a pair \((x, y)\):

• \(x\) represents the position along the horizontal axis.
• \(y\) represents the position along the vertical axis.

For a parabola's vertex, the coordinates \((h, k)\) tell us where the peak or the lowest point is situated. In our exercise, the coordinates \((3, 1)\) show that the vertex is at 3 units along the x-axis and 1 unit along the y-axis. Understanding these coordinates helps visualize the graph and analyze its properties.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's the foundation for expressing mathematical ideas, from basic equations to complex functions:

• In algebra, letters are often used to represent numbers that may vary.
• It allows us to solve for unknowns and understand relationships between variables.

In the context of our quadratic function, algebraic techniques help us manipulate the equation into different forms, such as vertex or standard form. Mastery of algebra is crucial for tackling more advanced math topics and solving real-world problems efficiently. Algebra transforms the way we approach problem-solving across various fields.