Problem 36 - Precalculus : Building Concepts and Connections

Question

Write each quadratic function in the form \(f(x)=a(x-h)^{2}+k\) by completing the square. Also find the vertex of the associated parabola and determine whether it is a maximum or minimum point. $$f(x)=-x^{2}-4 x+5$$

Step-by-Step Solution

Answer

The quadratic function in vertex form is \(f(x) = - (x + 2)^2 + 1\); the vertex of the parabola is (-2,1) and it's a maximum point.

Step 1: Write the quadratic function

First, write down the quadratic function given: \(f(x) = -x^2 - 4x + 5\)

Step 2: Group the terms

Next, group the \(x^2\) and \(x\) terms together, leaving the constant term separate: \(f(x) = - (x^2 + 4x) + 5\)

Step 3: Complete the square

The term to be added in the bracket to complete its square is \((b/2a)^2\). Here, \(a = -1\) and \(b = 4\), so add and subtract \((4/2 * -1)^2 = -4\) in the bracket and re-arrange: \(f(x) = - (x^2 + 4x - 4) -4 + 5 = - (x + 2)^2 + 1\)

Step 4: Write in vertex form

The transformed expression for the function \(f(x)\) is now in vertex form: \(f(x) = - (x + 2)^2 + 1\)

Step 5: Identify the vertex

The vertex of the parabola is at the point (h,k), where h and k are determined by the vertex form equation \(f(x) = a(x-h)^2 + k\). So, here the vertex is at (-2,1)

Step 6: The nature of the vertex

Since the coefficient of \(x^2\) in the original equation is negative, the parabola opens downwards. So, the vertex is a maximum point

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that makes it easier to analyze, specifically turning it into vertex form. This method involves creating a perfect square trinomial from the quadratic part of the function. For a quadratic equation given by

• \[f(x) = ax^2 + bx + c\]

The aim is to rewrite it as

• \[f(x) = a(x-h)^2 + k\]

Here, \( h \) and \( k \) are constants that indicate the vertex’s coordinates. Begin by factoring out the coefficient of the \(x^2\) term if it's not 1. Next, identify the value to complete the square inside the bracket, which is

• \(\left(\frac{b}{2a}\right)^2\)

Add and subtract this number inside the bracket to form a perfect square trinomial. Consequently, you can factor the expression as a square

• \((x-h)^2\)

This manipulation allows you to rewrite the equation in a more manageable form for further analysis.

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents the maximum or minimum value of the quadratic function, depending on the direction the parabola opens. For the equation written in vertex form

• \[f(x) = a(x-h)^2 + k\]

The vertex is given by the point

• \((h, k)\)

In the context of the provided equation,

• \(f(x) = - (x + 2)^2 + 1\)

We can identify the vertex as

• \((-2, 1)\)

Understanding the vertex enables us to identify key characteristics of the parabola, such as its axis of symmetry, which is the vertical line that passes through the vertex, and whether the parabola has a maximum or minimum point. If \(a\) is negative, the parabola opens downwards, and the vertex is a maximum point. Conversely, if \(a\) is positive, the parabola opens upwards, and the vertex is a minimum point.

Parabola Properties

A parabola is a symmetric curve that arises from quadratic functions. It has several properties that determine its shape and orientation.

• **Direction**: The sign of the quadratic coefficient \(a\) (in \(f(x) = ax^2 + bx + c\)) indicates if the parabola opens upwards (when \(a > 0\)) or downwards (when \(a < 0\)).
• **Vertex**: The vertex acts as either the highest or lowest point of the parabola depending on its direction.
• **Axis of Symmetry**: This is the vertical line that divides the parabola into two mirror-image halves. It can be expressed as \[x = h\]when the parabola is in vertex form \(y = a(x-h)^2 + k\).
• **Maximum or Minimum**: If the parabola opens upwards, the vertex represents the minimum point of the function. If it opens downwards, it represents the maximum.

Understanding these properties enables one to graph the function accurately and deduce several characteristics of its behavior without plotting numerous points.