Problem 14 - Algebra and Trigonometry

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$ f(x)=3 x^{2}-12 x+1 $$

Step-by-Step Solution

Answer

The coordinates of the vertex of the given parabola are (2, -3).

Step 1: Identify values of a, b and c

From the given quadratic function \(f(x) = 3x^2 - 12x + 1\), the values of a, b, and c are \(a = 3\), \(b = -12\), \(c = 1\).

Step 2: Find the x-coordinate of the vertex

To find the x-coordinate of the vertex, use the formula \(h = -b / (2a)\). Substituting \(a = 3\) and \(b = -12\) into this formula gives \(h = -(-12) / (2*3) = 2\). So, the x-coordinate of the vertex is 2.

Step 3: Find the y-coordinate of the vertex

Substitute the obtained x-coordinate \(h = 2\) into the function to find the y-coordinate \(k\). This gives \(k = f(2) = 3*(2)^2 - 12*2 + 1 = -3\). So, the y-coordinate of the vertex is -3.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial that can be described by an equation of the form \( f(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The graph of a quadratic function is called a parabola. It is a symmetrical curve that either opens upwards, when \(a > 0\), or downwards, when \(a < 0\).

The key properties of a parabola include its vertex, the highest or lowest point on the graph, axis of symmetry, a vertical line that passes through the vertex, and the direction of opening, determined by the sign of \(a\). In the exercise provided, to determine the coordinates of the vertex of the parabola for the quadratic function \(f(x) = 3x^2 - 12x + 1\), one must comprehend these characteristics as they come into play while calculating the vertex, which serves as the turning point of the parabola.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method enables calculating geometric figures' properties, such as points, lines, and curves, by turning them into algebraic problems.

In the context of parabolas and quadratic functions, coordinate geometry provides a way to locate the vertex's coordinates on the Cartesian plane. The vertex can be identified by the coordinates \( (h, k) \) where \( h \) is the x-value and \( k \) is the y-value. These values give us a specific point where the parabola changes direction. Using algebraic methods, such as the step-by-step solution for finding the vertex of a parabola provided in the exercise, one can find the exact location of the vertex on the graph. Understanding the relationship between algebraic expressions and geometric figures is essential in coordinate geometry and is exemplified when locating features on a parabola.

Completing the Square

Completing the square is an algebraic technique used to solve quadratic equations, derive the quadratic formula, and understand the properties of quadratic functions like the vertex form. To complete the square, one manipulates the equation so that the left-hand side forms a perfect square trinomial, which is an expression that can be factored into a square of a binomial.

Example

For the quadratic function \( f(x) = ax^2 + bx + c \), the process involves dividing \(b\) by \(2a\) and adding the square of this result to both sides of the equation to get a perfect square trinomial.

In the context of finding a vertex, this technique can simplify locating the x-coordinate by transforming the given quadratic function into vertex form. Once in vertex form, the coordinates of the vertex can be read directly from the equation, providing an alternative method to the one used in the solution of the given exercise. Although the exercise used a different approach, completing the square offers deeper insight into the properties of parabolas and their symmetrical nature.