Problem 8 - College Algebra

Question

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ f(x)=x^{2}-x $$

Step-by-Step Solution

Answer

The vertex form is \( f(x) = (x - \frac{1}{2})^2 - \frac{1}{4} \) with vertex \( (\frac{1}{2}, -\frac{1}{4}) \).

Step 1: Identify the Quadratic Function Components

The given function is \( f(x) = x^2 - x \). It is already in the quadratic form \( ax^2 + bx + c \) where \( a = 1 \), \( b = -1 \), and \( c = 0 \).

Step 2: Determine the Standard Form Formula

The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. We need to convert the given quadratic expression into this form.

Step 3: Use Completing the Square Method

To rewrite \( x^2 - x \) in vertex form, complete the square. Start with \( f(x) = x^2 - x \). The expression can be written as \( x^2 - x = (x^2 - x + \frac{1}{4}) - \frac{1}{4} \) to complete the square. This becomes \((x - \frac{1}{2})^2 - \frac{1}{4} \).

Step 4: Rewrite in Standard (Vertex) Form

The function in vertex form is \( f(x) = (x - \frac{1}{2})^2 - \frac{1}{4} \). This shows the transformation from the original quadratic equation to the vertex form.

Step 5: Identify the Vertex

In the vertex form \( f(x) = (x - \frac{1}{2})^2 - \frac{1}{4} \), the vertex \( (h, k) \) is \( (\frac{1}{2}, -\frac{1}{4}) \).

Key Concepts

Vertex Form

Vertex form is a way of expressing a quadratic equation to make it easier to identify important features, such as the vertex. The vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \]where

• \(a\) is the same coefficient from the standard form \( ax^2 + bx + c \)
• \((h, k)\) is the vertex of the parabola.

Being in vertex form helps quickly determine the vertex \((h, k)\), which is the peak or trough of the parabola, depending on whether it opens up or down (determined by the sign of \(a\)).
Using the vertex form is beneficial for graphing, as it directly provides the parabola's vertex, a key point that dictates the graph's shape and direction.

Completing the Square

Completing the square is a method used to convert a quadratic equation into its vertex form. To accomplish this:
First, start with the quadratic equation in standard form \( ax^2 + bx + c \). Here’s a simplified way to understand the process:

• Take the coefficient of \(x\), divide it by 2, and square it.
• Add and subtract this squared value inside the equation to create a perfect square trinomial.

For the given exercise \( f(x) = x^2 - x \):- Identify \(b = -1\), then divide by 2 to get \(-\frac{1}{2}\) and square it to obtain \( \frac{1}{4} \).- Add and subtract \( \frac{1}{4} \) inside the equation: \( x^2 - x + \left( \frac{1}{4} \right) - \left( \frac{1}{4} \right) \).
This transforms into \((x - \frac{1}{2})^2 - \frac{1}{4} \), simplifying to vertex form.

Quadratic Equation

A quadratic equation represents a second-degree polynomial function, generally written as:\[ ax^2 + bx + c = 0 \]where:

• \(a\), \(b\), and \(c\) are constants.
• \(a eq 0\) to ensure the equation is quadratic.

Quadratic equations describe parabolic shapes, which can either open upwards (when \(a > 0\))or downwards (when \(a < 0\)).
These equations can be solved using different methods, such as factoring, using the quadratic formula, or completing the square. Understanding how to manipulate them into different forms, like the vertex form, can provide deeper insights into their graphical representation and the significance of their coefficients.