Problem 15 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=(x+5)^{2}-8 $$

Step-by-Step Solution

Answer

The vertex is (-5, -8).

Step 1: Identify the Parabola in Vertex Form

The given function is \( f(x) = (x+5)^2 - 8 \). This function is in the vertex form of a parabola, \( f(x) = a(x-h)^2 + k \), where \( (h,k) \) is the vertex of the parabola.

Step 2: Extract the Vertex Coordinates

Compare the given function \( f(x) = (x+5)^2 - 8 \) with the general vertex form \( f(x) = a(x-h)^2 + k \). Here, \( h = -5 \) and \( k = -8 \).

Step 3: Write Down the Vertex

The vertex of the parabola is \( (h,k) = (-5, -8) \).

Key Concepts

Vertex Form

The vertex form of a quadratic function is incredibly useful for determining the key features of a parabola. It is written as: f(x) = a(x-h)^2 + k, where

• \( a \) determines the direction and the width of the parabola.
• \( h \) is the \( x \)-coordinate of the vertex.
• \( k \) is the \( y \)-coordinate of the vertex.

This form very clearly shows where the vertex is located, and it simplifies finding this point. For example, given the function \( f(x) = (x+5)^2 - 8 \), we can easily identify \( h = -5 \) and \( k = -8 \). Therefore, the vertex is at \( (-5, -8) \).

Parabola

A parabola is a U-shaped curve that can open up or down. In a quadratic function, the graph forms a parabola, and the position and shape of this parabola are determined by the equation's coefficients. The standard form of a quadratic equation is \( y = ax^2 + bx + c \), but the vertex form \( y = a(x-h)^2 + k \) is particularly helpful in quickly finding the vertex. Key features of a parabola:

• Vertex: The highest or lowest point on the parabola, depending on its direction.
• Axis of Symmetry: A vertical line that passes through the vertex dividing the parabola into two symmetric halves.
• Direction of Opening: If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.

For the function \( f(x) = (x+5)^2 - 8 \), the parabola opens upwards because the coefficient of the squared term is positive. The vertex is at \( (-5, -8) \), and the axis of symmetry is the line \( x = -5 \).

Quadratic Function

A quadratic function is a second-order polynomial function with the general form: \( f(x) = ax^2 + bx + c \), where:

• \( a \) is the coefficient of the squared term.
• \( b \) is the coefficient of the linear term.
• \( c \) is the constant term.

The graph of a quadratic function is a parabola. This type of function is fundamental in algebra and appears in various real-world scenarios such as physics, engineering, and economics. To convert a quadratic function from its standard form \( ax^2 + bx + c \) to its vertex form \( a(x-h)^2 + k \), one can complete the square. The vertex form simplifies many problems by making the vertex and transformations of the parabola directly visible. For example, starting with the standard form \( f(x) = x^2 + 10x + 17 \), completing the square yields the vertex form \( f(x) = (x+5)^2 - 8 \). This transformation shows the vertex \( (-5, -8) \).