Factored Form

The Factored Form is when a quadratic function is expressed in the form:

\(y = a(x-p)(x-q)\)

• \(x\) represents the independent variable
• \(y\) represents the dependent variable
• \(a\) represents the compression or stretch factor of the function and determines whether it opens upward or downward
• \(p\) and \(q\) represent the roots or \(x\)-intercepts of the function

You can identify the roots of the function when \(y\) is set to \(0\).

NOTE: The signs of the \(x\)-intercepts are opposite to what they are inside the function. For example, the x-intercepts of the function \(y = (x-3)(x+5)\) are \(3\) and \(-5\), NOT \(-3\) and \(5\).

A helpful trick is to set \(x\) and the corresponding value to opposite sides. Doing so will change the sign of the value.

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Converting Factored Form to Standard Form

Here are the steps required to convert an equation expressed in Factored Form to Standard Form:

1. In order to convert to Standard Form, you must first expand the Factored Form through distributive property (also known as the FOIL method)
2. Once everything is expanded and all like terms are collected, you must then multiply each term of the result by the value of a
3. Finally, rearrange the function if necessary so that the terms appear in descending order (i.e. \(x^2 + x + c\))

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Converting Factored Form to Vertex Form

Here are the steps required to convert an equation expressed in Factored Form to Vertex Form:

1. Convert to Standard Form following the steps above.
2. Then Complete the Square

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Graphing Factored Quadratic Functions

The main factors that are useful for sketching a parabola in Factored Form are:

• \(x\)-intercepts
• \(y\)-intercept
• vertex

Outlined below are the steps required to find the main parts of a parabola:

1. We can determine the \(x\)-intercepts of a function by setting \(x\) and the bracket values to opposite sides as outlined at the start of this lesson
2. In order to determine the \(y\)-intercept, all we need to do is set \(x = 0\), similar to finding the \(y\)-intercept of a linear function. Then, solve by multiplying the remaining values
3. In order to determine the vertex, we must first divide the sum of the \(x\)-intercepts by \(2\) to determine the \(x\)-coordinate of the vertex (also known as the Axis of Symmetry)
4. Next, we must place this \(x\)-value into the original function which will determine the \(y\)-coordinate of the vertex (also known as the Optimal Value)

Example

Identify the \(x\)-intercepts, \(y\)-intercept and Axis of Symmetry of the factored function \(y = (x - 3)(x + 2)\)

First, we can identify the \(x\)-intercepts by placing the \(x\)'s and values on opposite sides:

\(x = 3\) AND \(x = -2\)

Next, we can determine the \(y\)-intercept by placing \(0\) as the \(x\)-values. Then, solve:

\(y = (0 - 3)(0 + 2)\)

\(y = (-3)(2)\)

\(y = -6\)

Then, we can determine the\(x\)-coordinate (or Axis of Symmetry) of the vertex by dividing the sum of the \(x\)-coordinates by \(2\):

\(\text{AOS} = \cfrac{3 - 2}{2}\)

\(\text{AOS} = \cfrac{1}{2}\)

Finally, we can determine the \(y\)-coordinate (or Optimal Value) of the vertex by plugging the \(x\)-coordinate into the original function:

\(\text{OV} = \left(\cfrac{1}{2} - 3\right)\left(\cfrac{1}{2} + 2\right)\)

\(\text{OV} = \left(\cfrac{1}{2} - \cfrac{6}{2}\right)\left(\cfrac{1}{2} + \cfrac{4}{2}\right)\)

\(\text{OV} = \left(-\cfrac{5}{2}\right)\left(\cfrac{5}{2}\right)\)

\(\text{OV} = -\cfrac{25}{4}\)

Therefore, we can determine that the \(x\)-intercepts are \(\boldsymbol{3}\) and \(\boldsymbol{-2}\), the \(y\)-intercept is \(\boldsymbol{-6}\) and the vertex is \(\boldsymbol{\left(\cfrac{1}{2}, -\cfrac{25}{4}\right)}\).

x Values	-4	-3	-2	-1	0	1	2
y Values	-10	0	6	8	6	0	-10

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