Linear
pp 18-32 in workbook
Linear Equations:
point-slope form: y - y_1 = m(x - x_1)
standard form: ax + by = c
slope intercept form: y = mx +b
remember: (x_1 , y_1) are just the terms of ONE point on the line. If you're given two points, then you'll use (x_1, y_1) as one point, and (x , y) as your other.
Constant of Variation:
direct: y = kx
indirect: xy=k
Types of lines:
parallel lines have the same slope (m value)
perpendicular lines have slopes that are negative reciprocals of each other (ex. (1/2) and -2)
Systems of Equations:
manipulating equations to eliminate a variable when the two equations are added
substitution
Absolute Values:
terms inside | | can be + or - to satisfy the formula
ex. | x | = 2 --> x = 2 or x = -2
Inequalities:
***when you're multiplying by a negative number, the direction of the inequality symbol is reversed.
ex. -2x > 4 becomes x < -2
Functions
pp 11-17 in workbook
is it a function? vertical line test - if it fails the vertical line test, it's not a function
NO DIVISION BY 0
be able to identify the domain and range of functions (graphically or with coordinate points), find the roots, and identify the independent and dependent variables
evaluate a function f(x) at a that's saying, x = a, so you plug a into your equation and solve for f(a)
Sample Questions EOC
If you have questions on these notes, any problems in the workbook, or questions about problems for tomorrow (pp 1- and pp 7-10), comment here or bring your questions to class!
Tuesday, June 11, 2013
Friday, June 7, 2013
EOC Review
The top 5 areas of our focus during EOC Review 2013 are....
Linear Functions, Equations, and Inequalities
Characteristics and Behaviors of Functions
Arithmetic and Geometric Sequences
Quadratic Functions and Equations
Exponential Functions and Equations
I'm working on things to help you review/study for those areas specifically, but in the meantime,
Algebra IXL, this website has different quiz type questions that you can go through and get immediate feedback. It's also arranged by common core standard (really similar to what we use) so you can have some specific practice with whichever topic you're worried the most about.
Other EOC things:
Study Help
Calculator Policy
Have a great weekend!
Linear Functions, Equations, and Inequalities
Characteristics and Behaviors of Functions
Arithmetic and Geometric Sequences
Quadratic Functions and Equations
Exponential Functions and Equations
I'm working on things to help you review/study for those areas specifically, but in the meantime,
Algebra IXL, this website has different quiz type questions that you can go through and get immediate feedback. It's also arranged by common core standard (really similar to what we use) so you can have some specific practice with whichever topic you're worried the most about.
Other EOC things:
Study Help
Calculator Policy
Have a great weekend!
Wednesday, June 5, 2013
Quadratics (Completing the Square and our Wonderful Quadratic Equation)
Completing the Square:
I've included a link if you're wanting to go back and remember how completing the square works. Remember, we just want to make the right side of the equation a perfect square trinomial (which is in the form of ax^2 + 2kx + k^2 --- don't worry too much about the k's. It's just to show you that your b value for the quadratic is just 2k and your c value for the quadratic is just k^2.
The idea behind Completing the Square
Steps:
1. get the equation so that you have x^2 + 2kx on the right, and = - c on the left.
2. Find your b term
3. Divide your b term by 2
4. Got that answer? Square it.
5. That's now your new c value (which we can also label as k^2).
Once you have that new c value, you factor the trinomial into it's perfect square factored form, take the square root of both sides of your equation, and solve for x (remember, that's your root!) Don't forget when we take the square root of a number, it's going to be either that number, or the negative of that number. ((-4)(-4) = 16 OR (4)(4) = 16).
EXAMPLES!
Our general quadratic formula will be in the form:
I've included a link if you're wanting to go back and remember how completing the square works. Remember, we just want to make the right side of the equation a perfect square trinomial (which is in the form of ax^2 + 2kx + k^2 --- don't worry too much about the k's. It's just to show you that your b value for the quadratic is just 2k and your c value for the quadratic is just k^2.
The idea behind Completing the Square
Steps:
1. get the equation so that you have x^2 + 2kx on the right, and = - c on the left.
2. Find your b term
3. Divide your b term by 2
4. Got that answer? Square it.
5. That's now your new c value (which we can also label as k^2).
Once you have that new c value, you factor the trinomial into it's perfect square factored form, take the square root of both sides of your equation, and solve for x (remember, that's your root!) Don't forget when we take the square root of a number, it's going to be either that number, or the negative of that number. ((-4)(-4) = 16 OR (4)(4) = 16).
EXAMPLES!
x2 – 2x – 48 = 0
x2 – 2x – 48 + 48 = 0 + 48
x2 – 2x =
48
Our b term is -2.
Half of -2 is -1, then we square that, and get (-1)2 = 1
x2 – 2x + 1 = 48 + 1
x2 – 2x + 1 = 49
(x + 1)(x + 1) = 49
(x + 1)2 = 49
sqrt((x + 1)2) = +/- sqrt(49)
(x + 1) = +/- sqrt(49)
So we have two situations:
either….
(x + 1) = + 7 OR
(x + 1) = - 7
x = 6 OR x = -8
Quadratic Formula:
Remember, it might look scary, but this thing is going to be your new best friend.
ax2 + bx + c = 0
The a, b, and c values from your equation just replace their respective letters in the formula.
Example:
x2 - 2x - 1 = 0
a = 1
b = -2
c = -1
x = -(-2) +/- sqrt( (-2^2) - 4(1)(-1)) / 2(1)
(Sorry, blogger doesn't like it when I try and type these equations to look beautiful in word and copy them into here - sqrt( ) is just the square root of whatever is in the parentheses!)
x = 2 +/- sqrt( 4 + 4 ) / 2
x = 2 + sqrt(8) / 2 OR x = 2 - sqrt(8) / 2
(Aside: We can break 8 into 2*4, and 4 is a perfect square, so we can take the square root of the 4 and leave the left over 2 under the square root, so it'll look something like, 2sqrt(2))
Since we can pull a factor of 2 out of both terms in the numerator (with the little amount of math magic / root properties), we find:
x = 2(1 + sqrt(2) / 2 OR x = 2(1 - sqrt(2)) / 2
The twos in the numerator and denominator cancel out, leaving us with:
x = 1 + sqrt(2) OR x = 1 - sqrt(2)
TADA!
Links!
NOTE: I really did record myself singing the quadratic formula song, but it's not uploading at the moment. I will figure it out, though!
Tuesday, June 4, 2013
Quadratics (Monday and Tuesday lessons)
Manipulating the Parameters
Above is the website that we looked at on Friday to examine what effect the different variables in a quadratic equation have on its graph. Please look at this site so that you know how to predict the overall shape of the graphs!
I noticed a lot of confusion today in regards to factoring quadratics. Remember, a quadratic function is always in the form ax2 + bx + c = 0. All of the variables can be positive or negative, whole numbers or fractions, (they will always be whole numbers for this class), but our a value will never be 0. If our a term is 0, then we're left with bx + c = 0, which we recognize to be a linear function.
Because they will always be in this form (look familiar? check out the trinomials posts!) Always, always, ALWAYS check your equation first to see if you can pull out the GCF (Greatest Common Factor). Sometimes this GCF can be something subtle, like pulling out a -1, so make sure you check every time!
Once you have the GCF pulled out (if it's anything other than +1), to factor the quadratic you're just performing the same steps as you would when factoring a trinomial. If trinomials are still difficult, definitely check out the links for those pages - I tried to include both extra samples and a video example for each part!
Our roots for a quadratic equation are just our x-interepts, so when we're looking for the roots of an equation, you're going to figure out what the value(s) of x is(are) when y=0. In other words, we're trying to solve the equation for x, but we can either have 1 or 2 real number values that satisfies the equation. There is also the option of having imaginary roots, but since that is not something we need to worry about until Algebra 2, just assume I'm not mean enough to give you a problem you ABSOLUTELY CANNOT find the roots of (I promise I'm not). If you have a factored form, and it's set equal to zero, everything is ice from there.
Above is the website that we looked at on Friday to examine what effect the different variables in a quadratic equation have on its graph. Please look at this site so that you know how to predict the overall shape of the graphs!
I noticed a lot of confusion today in regards to factoring quadratics. Remember, a quadratic function is always in the form ax2 + bx + c = 0. All of the variables can be positive or negative, whole numbers or fractions, (they will always be whole numbers for this class), but our a value will never be 0. If our a term is 0, then we're left with bx + c = 0, which we recognize to be a linear function.
Because they will always be in this form (look familiar? check out the trinomials posts!) Always, always, ALWAYS check your equation first to see if you can pull out the GCF (Greatest Common Factor). Sometimes this GCF can be something subtle, like pulling out a -1, so make sure you check every time!
Once you have the GCF pulled out (if it's anything other than +1), to factor the quadratic you're just performing the same steps as you would when factoring a trinomial. If trinomials are still difficult, definitely check out the links for those pages - I tried to include both extra samples and a video example for each part!
Our roots for a quadratic equation are just our x-interepts, so when we're looking for the roots of an equation, you're going to figure out what the value(s) of x is(are) when y=0. In other words, we're trying to solve the equation for x, but we can either have 1 or 2 real number values that satisfies the equation. There is also the option of having imaginary roots, but since that is not something we need to worry about until Algebra 2, just assume I'm not mean enough to give you a problem you ABSOLUTELY CANNOT find the roots of (I promise I'm not). If you have a factored form, and it's set equal to zero, everything is ice from there.
(thing 1)*(thing 2) = 0 ---> implies that either (thing 1) = 0 OR (thing 2) = 0
With that in mind, hopefully the rest of the week will be easier to follow. We're going to go over completing the square again and do sample problems tomorrow in class.
REMEMBER: if you're stuck on something, you are always welcome to come in during lunch or after school to work one-on-one!!!
PS: I added this site to Mrs. Page's classroom webpage that you all already have access to. I'll make sure to double post any assignments or EOC help to ensure the maximum number of people
Monday, June 3, 2013
Trinomials, a>1
Since trinomials with a > 1 come up when we're finding the roots, I thought I'd offer an alternative way to factor these types of quadratic equations. It's definitely something we can talk about in class too.
3x2
- 8x + 4
You’re going
to start by multiplying your a and c terms.
This product will become your new c
term.
x2
- 8x + 4(3) = x2 - 8x + 12
Then factor x2
- 8x + 12
x2
- 8x + 12
 |
Using our
X-factor, we can see that -6 and -2 multiply to get 12, and add together to get
-8. Since this is a trinomial in which a
= 1, we can just insert the factors into their factored form:
(x – 6)(x –
2)
Our only
problem is that our -6 and -2 still have that 3 in them (from when we
multiplied our a term and our c term.
All we have to do is divide the 3 out of our factor terms:
-6/3 = -2 -2/3 doesn’t simplify - but that’s okay
(x – 2)
makes sense intuitively. That’s one
piece of our factored form.
To find the
other piece of our factored form, we have to simplify that fraction
(3x – 2)
(x - 2/3) to simplify the fraction, I simply multiply a
3 through…
(3x – 3(2/3)) = (3x – 2)
So now we
have our two factored pieces: (x – 2)(3x
– 2)
If you still
have your factoring polynomials guide, this is called the slip-and-slide method
for factoring trinomials.
Here's a video, too. The method we learned in class is what he refers to as the A.C. method...
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