Problem 16 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=-(x-2)^{2}+1 $$

Step-by-Step Solution

Answer

The vertex is \( (2, 1) \).

Step 1: Understand the Vertex Form

The vertex form of a quadratic function is given by \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

Step 2: Identify Coefficients

Compare the given function \( f(x) = -(x-2)^2 + 1 \) to the vertex form. Here, \( a = -1 \), \( h = 2 \), and \( k = 1 \).

Step 3: Determine the Vertex

The vertex \( (h, k) \) of the parabola is \( (2, 1) \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function of degree 2. It can be written in the general form: \(f(x) = ax^2 + bx + c\). Here,

• 'a', 'b', and 'c' are coefficients.
• The graph of a quadratic function is a U-shaped curve called a parabola.

The coefficient 'a' determines the direction in which the parabola opens:

• If 'a' > 0, the parabola opens upward.
• If 'a' < 0, the parabola opens downward.

Each point on the graph represents a value that the function can take. The highest or lowest point on the graph is known as the vertex. Understanding the quadratic function’s structure helps in analyzing its properties, such as the vertex, axis of symmetry, and the direction of its opening.

Vertex Form

The vertex form of a quadratic function gives us immediate information about the vertex of the parabola. It is written as: \(f(x) = a(x-h)^2 + k\). Here,

• 'a' is the same coefficient as in the standard form and affects the parabola's direction.
• '(h, k)' is the vertex of the parabola.

Identifying the vertex is straightforward with this form. For example, if we have \(f(x) = -(x-2)^2 + 1\), we see ‘a’ is -1, 'h' is 2, and 'k' is 1. Thus, the vertex of this particular quadratic function is at (2, 1). The vertex form is particularly useful for graphing parabolas and analyzing their features without needing to complete the square.

Parabola

A parabola is the graph of a quadratic function and has a distinctive U-shape. Some key features include:

• The vertex: It is the highest or lowest point of the parabola.
• The axis of symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
• The direction: Determined by the sign of coefficient 'a'. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.

Consider the function \(f(x) = -(x-2)^2 + 1\): Here, the vertex is at (2, 1), and the parabola opens downward because ‘a’ is -1. By understanding these characteristics, you can sketch the parabola and interpret the graph's behavior easily.