Unit 5 (February)
Adding and subtracting polynomials: To add or subtract polynomials, you simply add or subtract the like terms (remember 7th grade????).
Link for adding and subtracting help
Link for adding and subtracting help
Multiplying polynomials: Previously, you have distributed single terms (monomials) to multi-termed expressions. For example, 2(x + 4) = 2x + 8 and 2x(3y + 6) = 6xy + 12x. To multiply polynomials you are essentially doing to same thing. Mathematically speaking, you are simply distributing each term of one polynomial to each term of the other. For example, (x + 2)(x + 3) = x^2 + 3x + 2x + 6..... Your text, all online help sites, and every math teacher you will ever meet (ever) will stress the mnemonic FOIL (first outer inner last) to multiply binomials (multiply the first terms of each expression, then the outer 2, then the inner 2, and finally the last 2). However, this is only applicable when multiplying binomials. I would prefer if you remember the fact that you are simply distributing one polynomial into the other. With this in mind, you will be able to multiply any polynomial (for example, (x^2 + 3x +7)(x^3 + 4x^2 + 5x + 10) ).
Video example
Solving polynomials in factored form: Solving polynomials in factored form is quite simple. If I said to solve (x +2)(x - 5) = 0 for x, your first thought might be to FOIL the two and get x^2 -3x - 10=0. However, we do not yet know the steps to solve this form of a polynomial.
Therefore, think of it this way...... If I had the equation A x B = 0, what does this mean about A or B? One of them has to be 0 because 0 times anything = 0. Therefore, when given (x +2)(x - 5) = 0 , you have two things being multiplied and equaling 0. What number would make the first expression = 0? What number would make the second expression = 0? -2 and +5.
Therefore, x = -2 or x = 5.
Video help
Graphing polynomials (quadratics specifically) in factored form: If asked to graph y = (x + 2)( x - 4), we can easily graph this equation despite it not being linear. First of all, we know that this will FOIL as x^2 -2x - 8, and is therefore a quadratic. Quadratics form parabolas (giant U's), that are symmetrical (the left side is symmetrical to the right).
x intercepts: We also know how to find the x-intercepts of an equation (when y = 0) and the y-intercepts of an equation (when y = 0).
y-intercept(s): These occur when an equation's x=0
Axis of symmetry: This is simply the line of symmetry that splits the parabola down the middle. This is easy to find, because it will lie directly in between the 2 x-intercepts.
Vertex: This is the maximum or minimum point of the graph. If the parabola opens downward, it will have a maximum, if it opens upward (like below) it will have a minimum. This point is also easy to find because the axis of symmetry goes directly through the vertex.
Factoring simple polynomials: To "factor" a polynomial or expression, you are breaking it down into a product of its factors. For example, If I said 'factor the number 8' you would break it down into 2x4. Since 1 is a factor of EVERYTHING, you would not say 1x8 is the "correct" factoring (it is assumed knowledge that you know 1x8 = 8) To factor a simple polynomial, like 2x + 6, you simply "undistribute" the greatest common factor of each term. Or try thinking of it this way.... If you had the expression 2(x + 3), you would distribute or multiply the 2 to the x and 3. Factoring is the opposite. So if I gave you 2x + 6, determine what each term was multiplied by to get that polynomial. In this case, 2 goes into both 2x and 6, so 2 is the greatest common factor. 2x + 6 would factor into 2(x + 3).
Some expressions are a little more difficult, but again, you are simply 'undistributing' or dividing out the greatest factor that each part has in common with each other. For example:
Factoring quadratics when a=1: Recall that factors are numbers or terms that go evenly into another number or term. For example, 2 and 4 go evenly into 8, and 2 and x go evenly into 2x. If a number/term has no factors besides 1 and itself, it is considered 'prime' (like 7, 13, x+1...). When factoring polynomials, I want you to channel your methods used to multiply them. Because essentially you are 'undoing' or doing the reverse of multiplying polynomials.
For example, when we multiply (x+4)(x+2) we get x^2 + 6x + 8. Using FOIL, First we get x^2, for Outer and Inner we get 6x, and Last we get 8. Using that knowledge, if I gave you x^2 + 6x + 8 and asked you what factors were multiplied to get this polynomial, you know that your FIRST terms gave you x^2 and the LAST terms multiplied to give you 8. You also know that when you multiply the OUTER and INNER terms they must add up to the LAST product as well. See the color coded example below:
Notice how the O-I in FOIL must add up to 6, but multiply to 8.
The leading coefficient (the coefficient of the squared x) is 1. Therefore, it has little bearing on the factoring. You simply look for factors of 8 that add up to 6. However, the 'true rule' is slightly different. ( I go over this further in a later lesson)
The leading coefficient (the coefficient of the squared x) is 1. Therefore, it has little bearing on the factoring. You simply look for factors of 8 that add up to 6. However, the 'true rule' is slightly different. ( I go over this further in a later lesson)
Factoring by grouping: Recall when factoring a simple polynomial that you simply pull out (or 'undistribute') the greatest common factor. When factoring by grouping, you are doing the same thing, however, the greatest common factor is not simply 1 term (like 4 or 2xy) but a 2 term expression (like x + 4). For example, suppose you had the following:
With this method of factoring (and your knowledge of factoring simple polynomials), you can factor even termed polynomials by grouping the terms into smaller sections, and pulling out their common factors. See the 2 examples below:
Factoring quadratics with a leading coefficient greater than 1 by implementing the grouping method: As discussed previously, the actual rule for factoring a quadratic in the form ax^2 + bx + c is not simply factors of c that add up to b. That would mean a has no effect on the factoring whatsoever (which is entirely wrong). The actual rule states that you want factors of the product of the leading coefficient and your constant that add up to the middle coefficient. In simpler terms, factors of a and c that add up to b. See below:
Factoring Quadratics completely: Above I described 4 different methods for factoring. However, the biggest issue will not be how to perform each method, rather, it will be deciding what method works best when given a random polynomial. There are a few tricks to determining this however....
Example 1: (ps, this is probably the 'most difficult' factoring you will have to perform at an Algebra 1 level)
Example 2:
Example 3:
Unit 6
Locus, Focus, and Directrix: Geometrically speaking, a parabola is a unique figure. A parabola is actually a collection of points that are equally distant from a given point and line. The "collection of points" is called a locus, the given point is the focus or focal point, and the given line is called the directrix. The parabola always opens towards the focus, and away from the directrix.
As we learned in class, if given a quadratic equation the focus can be found by discovering the point ( aos, 1/(4a)). Where your x-value is the axis of symmetry (just like the vertex's x-value) and the y value is 1/(4a).
Graphing simple quadratics in Standard Form: Recall that we can graph parabolas in factored form rather easily by simply finding the roots. We can then find the axis of symmetry and symmetry based on these roots. However, not all quadratics will be in factored form (if you're lucky they will be), in fact MANY quadratics will actually be PRIME. Therefore, we need to know a few basic principles of graphing quadratics before introducing Vertex form.
As you can see, by adding or subtracting a 'c' value, my parent function moves vertically up or down. Think about it......If your parent function is the red parabola above, then your blue parabola is adding one to every point. Therefore, your blue parabola will be exactly the same as your red parabola except for a vertical shift of +1.
The video below focuses on simple word problems involving standard form:
Solving Quadratic Equations by taking square roots:
So far we have discussed solving quadratic equations by factoring and using the zero-product property. We will now move to a few more "algebraic methods. First on our list is solving by taking square roots.
In order to take a square root, we need to ensure that the term containing the variable and exponent is by itself (isolate the quadratic term). Why? Because we do not have a "rule" for taking square roots of a sum or difference. Therefore, we can only take a square root (or a 3rd root, 4th root, etc...) when this term is by itself.
Secondly, this method is only useful if your quadratic term is the ONLY term containing a variable. Meaning, if you have 2x^2 = 10 then you can solve it by taking square roots. However, if your equation is 2x^2 + 5x = 10, then you cannot take square roots because you have another variable term to deal with and taking a square root will only complicate your problem.
Take a look at the video below:
In order to take a square root, we need to ensure that the term containing the variable and exponent is by itself (isolate the quadratic term). Why? Because we do not have a "rule" for taking square roots of a sum or difference. Therefore, we can only take a square root (or a 3rd root, 4th root, etc...) when this term is by itself.
Secondly, this method is only useful if your quadratic term is the ONLY term containing a variable. Meaning, if you have 2x^2 = 10 then you can solve it by taking square roots. However, if your equation is 2x^2 + 5x = 10, then you cannot take square roots because you have another variable term to deal with and taking a square root will only complicate your problem.
Take a look at the video below:
Deriving the Vertex form of a parabola:
Next let's look at how changing the x-value itself alters the parent function:
Take a look at the green equation, then find the root of your equation (the root occurs when y=0). Your root happens to be -2, therefore your vertex of the green equation is at (-2,0). Likewise, the root of your orange equation is at (2,0). It would appear then that by "adding" to the x-value (as is the case with the green equation) that you are moving your parent function left. Likewise, by subtracting from your x-value you are actually moving your equation right.
Now lets bring it all together:
- By adding or subtracting from your x-value directly you are translating or shifting your parabola left and right.
- By adding or subtracting from the parent function directly you are translating or shift your parabola up and down.
- By multiplying your function you are increasing or decreasing the width of your parabola. You can also change the direction of your parabola's opening.
Thus we have a new formula that appears very similar to our absolute value formula learned earlier this year:
Now lets bring it all together:
- By adding or subtracting from your x-value directly you are translating or shifting your parabola left and right.
- By adding or subtracting from the parent function directly you are translating or shift your parabola up and down.
- By multiplying your function you are increasing or decreasing the width of your parabola. You can also change the direction of your parabola's opening.
Thus we have a new formula that appears very similar to our absolute value formula learned earlier this year:
Vertex form allows us a second way to graph quadratic equations (recall that we already learned how to graph quadratics in factored form).
This second method comes in handy because, as you know by now, not all quadratics are factorable. Sometimes you have a prime quadratic that cannot be factored, therefore you are not able to graph it using the factored form.
Therein lies a small problem. In order to graph a quadratic in factored form you must first be able to convert it to factored form. The work here can be tricky so bear with me.
Converting quadratic equations into Vertex Form: Take a look at each quadratic equation below in Vertex Form, then if you multiply your squared portion out and simplify your like terms, take a look at the same equation written in Standard Form:
Notice how the Standard form for each is PRIME, however, somehow we were able to write "part" of the equation in a sort of factored out perfect square manner in order to write it as vertex form. Mathematically, the Vertex form is NOT factored, but it has something in it that appears factored. Each Vertex form uses or implements a perfect square. Therefore, the question becomes how exactly do we take a PRIME quadratic and convert it into something that looks squared? The answer is simple, we force it to be a perfect square: :)
The quadratic equation above was actually quite simple because my b-value was 4. I know that anytime my b-value is even there is a corresponding "perfect square" quadratic that has the same b-value. However, what about when b is not even?
To convert a quadratic equation that has an odd 'b' value we must first realize a few things:
To convert a quadratic equation that has an odd 'b' value we must first realize a few things:
What does this all mean? Well if you know the b value, then in order to create a "perfect square" your c value must be the square of half of your b value. For example:
Graphing Quadratics Summary (the Preferred Method for graphing): Based on the above concepts, you are now able to graph any quadratic equation no matter how it is written or presented. You have the option of choosing between graphing via factored form, simple standard form, vertex form, or converting a prime quadratic into vertex form.
See the examples below:
Graphing Quadratics Summary (the Preferred Method for graphing): Based on the above concepts, you are now able to graph any quadratic equation no matter how it is written or presented. You have the option of choosing between graphing via factored form, simple standard form, vertex form, or converting a prime quadratic into vertex form.
See the examples below: