The factored form is when a quadratic function is expressed in the form:

- \(x\) 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 \(0\).

Take note that 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.

Identify the x-intercepts of the function \(y = 2(x + 10)(x + 4)\)

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

Convert the quadratic function \(y = 2(x-8)(2x+3)\) from Quadratic Form to Standard Form

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

The main factors that are useful for sketching a parabola in factored form are:

- x-intercepts
- y-intercept
- vertex

- 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
- 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
- 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)
- 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)

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:

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

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

\(y = -6\)

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

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

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

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

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

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

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

Using the following table of values, graph the parabola, determine its x-intercepts, y-intercept and vertex, then determine its factored quadratic function

X | -4 | -3 | -2 | -1 | 0 | 1 | 2 |
---|---|---|---|---|---|---|---|

Y | -10 | 0 | 6 | 8 | 6 | 0 | -10 |

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