Week of March 12, 2018

Algebra 1

Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution.

The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical coefficients" of the quadratic equation they've given you to solve

The Quadratic Formula is derived from the process of completing the square, and is formally stated as:

The Quadratic Formula: For ax2 + bx + c = 0, the values of x which are the solutions of the equation are given by:

$x = \dfrac{-b \pm\sqrt{b^2 - 4ac\,}}{2a}$

For the Quadratic Formula to work, you must have your equation arranged in the form "(quadratic) = 0". Also, the "2a" in the denominator of the Formula is underneath everything above, not just the square root. And it's a "2a" under there, not just a plain "2". Make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back in" on your test, and you'll mess yourself up. Remember that "b2" means "the square of ALL of b, including its sign", so don't leave b2 being negative, even if b is negative, because the square of a negative is a positive.

In other words, don't be sloppy and don't try to take shortcuts, because it will only hurt you in the long run. Trust me on this!

Here are some examples of how the Quadratic Formula works:

• Solve x2 + 3x – 4 = 0

This quadratic happens to factor:

x2 + 3x – 4 = (x + 4)(x – 1) = 0

...so I already know that the solutions are x = –4 and x = 1. How would my solution look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, my solution looks like this:

$x = \dfrac{-(3) \pm \sqrt{(3)^2 - 4(1)(-4)\,}}{2(1)}$

$= \dfrac{-3 \pm \sqrt{9 + 16\,}}{2} = \dfrac{-3 \pm \sqrt{25\,}}{2}$

$= \dfrac{-3 \pm 5}{2} = \dfrac{-3 - 5}{2},\, \dfrac{-3 + 5}{2}$

$= \dfrac{-8}{2},\, \dfrac{2}{2} = -4,\, 1$

Then, as expected, the solution is x = –4, x = 1.

http://www.purplemath.com/modules/quadform.htm

Algebra 2

End Behavior of a Function

The end behavior of a polynomial function is the behavior of the graph of f(x)$f\left(x\right)$ as x$x$ approaches positive infinity or negative infinity.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

Degree	Leading Coefficient	End behavior of the function	Graph of the function
Even	Positive	f(x)→+∞, as x→−∞f(x)→+∞, as x→+∞$\begin{array}{l}f\left(x\right)\to +\infty ,\text{ as }x\to -\infty \\ f\left(x\right)\to +\infty ,\text{ as }x\to +\infty \end{array}$	Example: f(x)=x2$f\left(x\right)=x^2$
Even	Negative	f(x)→−∞, as x→−∞f(x)→−∞, as x→+∞$\begin{array}{l}f\left(x\right)\to -\infty ,\text{ as }x\to -\infty \\ f\left(x\right)\to -\infty ,\text{ as }x\to +\infty \end{array}$	Example: f(x)=−x2$f\left(x\right)=-x^2$
Odd	Positive	f(x)→−∞, as x→−∞f(x)→+∞, as x→+∞$\begin{array}{l}f\left(x\right)\to -\infty ,\text{ as }x\to -\infty \\ f\left(x\right)\to +\infty ,\text{ as }x\to +\infty \end{array}$	Example: f(x)=x3$f\left(x\right)=x^3$
Odd	Negative	f(x)→+∞, as x→−∞f(x)→−∞, as x→+∞$\begin{array}{l}f\left(x\right)\to +\infty ,\text{ as }x\to -\infty \\ f\left(x\right)\to -\infty ,\text{ as }x\to +\infty \end{array}$	Example: f(x)=−x3$f\left(x\right)=-x^3$

To predict the end-behavior of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative.

Example:

Find the end behavior of the function x4−4x3+3x+25$x^4-4x^3+3x+25$.

The degree of the function is even and the leading coefficient is positive. So, the end behavior is:

f(x)→+∞, as x→−∞f(x)→+∞, as x→+∞$\begin{array}{l}f\left(x\right)\to +\infty ,\text{ as }x\to -\infty \\ f\left(x\right)\to +\infty ,\text{ as }x\to +\infty \end{array}$

The graph looks as follows:

 https://www.varsitytutors.com/hotmath/hotmath_help/topics/end-behavior-of-a-function

Week of March 5, 2018

Algebra 1

Completing the Square

"Completing the Square" is where we ... 

... take a Quadratic Equation like this:		and turn it into this:
ax2 + bx + c = 0		a(x+d)2 + e = 0

For those of you in a hurry, I can tell you that:d = b2a

and:e = c − b²4a

But if you have time, let me show you how to "Complete the Square" yourself.

Completing the Square

Say we have a simple expression like x² + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x² + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2)²

In Algebra it looks like this:

x2 + bx	+ (b/2)2	=	(x+b/2)2
	"Complete the Square"

So, by adding (b/2)² we can complete the square.

And (x+b/2)² has x only once, which is easier to use.

Keeping the Balance

Now ... we can't just add (b/2)² without also subtracting it too! Otherwise the whole value changes.

So let's see how to do it properly with an example:

Start with:
	("b" is 6 in this case)
Complete the Square:
	Also subtract the new term
Simplify it and we are done.

The result:

x² + 6x + 7   =   (x+3)² − 2

And now x only appears once, and our job is done!

A Shortcut Approach

Here is a quick way to get an answer you may like.

First think about the result we want: (x+d)² + e

After expanding (x+d)² that becomes: x² + 2dx + d² + e

Now see if we can turn our example into x² + 2dx + d² + e form to discover d and e

Example: try to fit x² + 6x + 7 into x² + 2dx + d² + e

Now we can "force" an answer:

• We know that 6x must end up as 2dx, so d must be 3
• Next we see that 7 must become d² + e = 9 + e, so e must be −2

And we get the same result (x+3)² − 2 as above!

https://www.mathsisfun.com/algebra/completing-square.html


Algebra 2

Factoring and Graphing PolynomialsMathBitsNotebook.com Topical Outline | Algebra 1 Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

The skill of factoring is a helpful tool when graphing polynomial functions.
(In Algebra 1, PARCC is limiting this task to quadratic and cubic polynomial functions with linear and quadratic factors.)

A polynomial function is a function which is defined by a polynomial. (In plain English, it's a polynomial expression set equal to f(x), or some other notation.) Examples:   f(x) = x2 + x - 6;      P(x) = x3 - x2 - 12x;      y1 = x2 + 4x + 4

Basic information about the graphs of polynomial functions:

1. Continuous: The graphs of polynomial functions are continuous. You can draw the entire function without lifting your pencil from your paper.

2. Smooth: The graphs of polynomial functions are always nice smooth graphs.  The graphs do not contain holes, breaks or sharp point corners.

3. Turning Points: The "hills" or "valleys" where the graph changes direction from increasing to decreasing, or from decreasing to increasing, are often called turning points.  The number of turning points is never more than the degree of the polynomial minus one. For example, the graph of a 3rd degree polynomial function can have 2 turning points or fewer. [If the degree is n, the number of turning points is at most n - 1. There could be fewer.] Other terms for turning points can be relative maximum (or minimum) and local maximum(or minimum). The designation of "relative" (or "local") tells you that while this point may be a turning point, it may not be the maximum (or mininum) value reached by this function. It is only a maximim (or minimum) "relative" to a small section of the function surrounding this point. The one, true, largest (or smallest) value reached by the entire function is called the absolute maximum (or minimum), or the global maximum (or minimum). FYI: The plural of maximum is maxima. The plural of minimum is minima. Discussions of maxima and minima are referred to as extrema.

Here's where the factoring connection comes in to play:

4. Roots (or Zeros): If you plug in r (some real number) for x in a polynomial function, P(x), and get an answer of 0, the number, r, is called a root, or zero, of the polynomial. It can be quite tedious to repeatedly test random numbers looking for those numbers that give an answer of zero. A faster method for finding these roots (or zeros), is to factor the polynomial, and then set the factors equal to zero (the desired answer). This factoring method for finding roots, or zeros, utilizes thezero factor principle which states that "if a • b = 0, then either a = 0 and/or b = 0."  The real numbers that create the roots (or zeros) of a polynomial correspond to the x-intercepts of the graph of the polynomial function. This is valuable information when it comes to creating the graph of a polynomial (without a graphing calculator). It is also valuable if you are given the graph and are attempting to create a possible equation.  Note: A polynomial of degree 2 will have two roots (zeros), a polynomial of degree 3 will have three roots (zeros), and so on.

Example:

What happens when a factor repeats?

5. Multiplicity of Roots (or Zeros): We have seen in section 4 that the roots (zeros) of a polynomial correspond with the x-intercepts of the polynomial graph. In some situations, the graph will "cross" the x-axis at these points. In other situations, the graph may simply "touch" (be tangent to) the x-axis at these points. Let's see if we can determine, before we draw the graph, whether it will "cross" the x-axis at each root, or simply "touch" (be tangent to) the x-axis at each root. Consider the example at the right. The polynomial is of degree two, so there will be two roots (zeroes). The factor of (x + 3) is repeated twice, and can also be written as (x + 3)2. The number of times a factor appears in a polynomial is referred to as its multiplicity. In the example at the right, the factor (x + 3) has a multiplicity of 2, since it appears twice. It creates a "repeated root".  Multiplicity EVEN: When the multiplicity (the number of times a factor repeats) is two (or even), the graph will just "touch" (be tangent to) the x-axis at that point. Why? "EVEN" multiplicities are factors that occur an even number of times, and form squares. Since squares are always positive, the graph near the root (zero) will not change signs from positive (above the x-axis) to negative (below the x-axis), or vice versa. The graph will "touch" the x-axis at the root (zero) but remain on the same side of the x-axis.

Example degree 2:

What if the multiplicity is ODD? Consider the example at the right. The polynomial is of degree three, so there will be three roots (zeros). The factor of (x - 1) appears three times, and can be written as (x - 1)3.  Multiplicity ODD: When the multiplicity (the number of times a factor repeats) is three (or odd), the graph will "cross" the x-axis at that point. Did you notice?  If you see a factor such as (x - 1)3, the multiplicity is 3. If you see a factor such as (x + 2)2, the multiplicity is 2. If you see a factor such as (x + 3), the multiplicity is 1.  Note: When you factor a polynomial, the sum of the multiplicities equals the degree of the polynomial. x3 + x2 - 5x + 3 = (x - 1)2(x + 3) = (x - 1)2(x + 3)1 degree = 3 sum of multiplicities = 2 + 1 = 3

Example degree 3:

There is one more thing we can find out before we graph!

6. End Behavior: Looking at a few aspects of a polynomial will tell us what is happening at either end of the graph. We will be looking at the DEGREE of the polynomial and the SIGN of the leading coefficient to determine what is happening to the graph. End behavior refers to the appearance of a graph as it is followed indefinitely in either direction.
Even Degree Polynomial ("ends" behave similar to a quadratic) Leading coefficient POSITIVE: both "ends" are UP. Leading coefficient NEGATIVE: both "ends" are DOWN.	Odd Degree Polynomial ("ends" behave similar to a cubic) Leading coefficient POSITIVE: left end is DOWN and right end is UP. Leading coefficient NEGATIVE: left end is UP and right end is DOWN. https://mathbitsnotebook.com/Algebra1/Factoring/FCGraphing.html