Hey Gummy Bears,
Ever wonder where people are when they check out our blog?? Now you can know. I've put a visitor map at the bottom of the home screen. Let's see how many different countries are represented by the end of the semester.
Ms. Armstrong
Welcome to a world where creativity, ingenuity, and mathematics mesh together to create Grade 10 Precalculus math.
posted by abbas at 10:17 p.m.
The other format for straight-line equations is called the "point-slope" form. For this one, they give you a point (x1, y1) and a slope m, and have you plug it into this formula:
y – y1 = m(x – x1)
the subscripts are just intended to indicate the point they give you. You have the "x" and "y" that are always in your equation, and then you have the specific x and y from the point they gave you. Here's how you use the point-slope formula:
ex:
Find the equation of the straight line that has slope m = 4 and passes throughthe point (–1, –6).
They've given me m = 4, x1 = –1, and y1 = –6. I'll plug these values into the point-slope form:
y – y1 = m(x – x1)
y – (–6) = (4)(x – (–1))
y + 6 = 4(x + 1)
y + 6 = 4x + 4
y = 4x + 4 – 6
y = 4x – 2
You can find the straight-line equation using the point-slope form if they just give you a couple points:
Find the equation of the line that passes through the points (–2, 4) and (1, 2).
Then I can use either point as my (x1, y1), along with this slope Ive just calculated, and plug in to the point-slope form. Using (–2, 4) as the (x1, y1), I get:
y – y1 = m(x – x1)
y – (4) = (– 2/3)(x – (–2))
y – 4 = (– 2/3)(x + 2)
y – 4 = (– 2/3)x – 4/3
y = (– 2/3)x – 4/3 + 4
y = (– 2/3)x – 4/3 + 12/3
y = (– 2/3)x + 8/3
posted by Anonymous at 3:38 p.m.
March 15, 2007
HELLO! =)
the things we did today:
1. we checked whether anyone's having a hard time answering any number in exercise 14.
2. ms. armstrong told us that the analytic geometry test will be on monday, not tomorrow. tomorrow will be our work period.
3. ms. armstrong taught us how to determine the equations of lines; standard form or slope-intercept form.
First of all, standard form looks like this, Ax + By + C = 0. So if you see an equation that's in this format, it is in standard form. always remember that a, b, and c are integers. they can never be fractions. Then, slope-intercept form looks like this, y = mx + b. It is okay if you have m or b as fractions.
Now that you know that, you can now find the standard form or slope-intercept form even if the only information given are the 2 points.
for example:
Find the standard form of the equation of a line containing (6,1) and (-4,3)
first you ask yourself, what are you asked to find? in this equation, you are asked to find for the standard form of the equation. (Ax + By + C = 0)
second, what info are you given? the 2 points : (6,1) and (-4,3)
third, how can you use that info? since you have 2 points, you can try finding the slope.
m = y2 - y1 / x2 - x1
m = 3-1 / -4-3
m = 2 /-1
m = -1/5
now, that you've find out the slope, what equation can you use using a slope and a point?
POINT-SLOPE EQUATION. slope = -1/5 , points = (6,1) or (-4,3) - just chose between the 2 points. and then just substitute these numbers into the formula.
(y-y1) = m(x-x1) - formula of point-slope equation
y-1 = -1/5 (x-6)
--in this part, the first thing that comes into your mind is to just multiply -1/5 to x-6. But if you do that you'll have a fraction and remember, you can't have a fraction in a standard form. So what you do is, just multiply 5 to both sides so that you can eliminate the fraction -1/5
5(y-1) = (-1/5 (x-6) )5
5y-5 = -x+6
x+5y-5-6 = 0
x+5y-11 = 0 - i transfered everything to the other side because remember the standard
form should look like this (Ax + By + C = 0). Also, always remember that
the x should be always positive. so if the x becomes negative, just transfer
it to the other side along with the other numbers.
the answer is x + 5y - 11 = 0.
So what if you're asked to find for the slope-intercept form and you're given 2 points? you just have to do the same steps. find for the slope and do the point-slope equation. and you can determine the equation of the line already.
I think that's it. Also, don't forget to do Exercises 15, and 16.
I really hope i have helped you in some way!
the next one blogging is..(drumroll please)....JENNY! =)
HELLO! =)
the things we did today:
1. we checked whether anyone's having a hard time answering any number in exercise 14.
2. ms. armstrong told us that the analytic geometry test will be on monday, not tomorrow. tomorrow will be our work period.
3. ms. armstrong taught us how to determine the equations of lines; standard form or slope-intercept form.
First of all, standard form looks like this, Ax + By + C = 0. So if you see an equation that's in this format, it is in standard form. always remember that a, b, and c are integers. they can never be fractions. Then, slope-intercept form looks like this, y = mx + b. It is okay if you have m or b as fractions.
Now that you know that, you can now find the standard form or slope-intercept form even if the only information given are the 2 points.
for example:
Find the standard form of the equation of a line containing (6,1) and (-4,3)
first you ask yourself, what are you asked to find? in this equation, you are asked to find for the standard form of the equation. (Ax + By + C = 0)
second, what info are you given? the 2 points : (6,1) and (-4,3)
third, how can you use that info? since you have 2 points, you can try finding the slope.
m = y2 - y1 / x2 - x1
m = 3-1 / -4-3
m = 2 /-1
m = -1/5
now, that you've find out the slope, what equation can you use using a slope and a point?
POINT-SLOPE EQUATION. slope = -1/5 , points = (6,1) or (-4,3) - just chose between the 2 points. and then just substitute these numbers into the formula.
(y-y1) = m(x-x1) - formula of point-slope equation
y-1 = -1/5 (x-6)
--in this part, the first thing that comes into your mind is to just multiply -1/5 to x-6. But if you do that you'll have a fraction and remember, you can't have a fraction in a standard form. So what you do is, just multiply 5 to both sides so that you can eliminate the fraction -1/5
5(y-1) = (-1/5 (x-6) )5
5y-5 = -x+6
x+5y-5-6 = 0
x+5y-11 = 0 - i transfered everything to the other side because remember the standard
form should look like this (Ax + By + C = 0). Also, always remember that
the x should be always positive. so if the x becomes negative, just transfer
it to the other side along with the other numbers.
the answer is x + 5y - 11 = 0.
So what if you're asked to find for the slope-intercept form and you're given 2 points? you just have to do the same steps. find for the slope and do the point-slope equation. and you can determine the equation of the line already.
I think that's it. Also, don't forget to do Exercises 15, and 16.
I really hope i have helped you in some way!
the next one blogging is..(drumroll please)....JENNY! =)
Labels: analytic geometry, janelle, scribe
Thursday, March 08, 2007
posted by AngelaG. at 6:13 p.m.
There are Three different ways to graph lines. We already learned the Slope Method. Then today we learned the Intercept Method and the Slope Intercept method.
First, the intercepts method.
(By the way, you know what intercept means right? Well, if not, it means "to run into" or "block" )
And thats what the intercepts method means. You look for the two points where the line crosses the X axis, and the Y axis.
Once you find the line, you'll find your coordinates.
A question can look like this = "3x + 4y - 12 = 0"
To find your coordinates, you want to find the value of X, then the value of Y.
To find the coordinates, first you wanna find the X axis. to find it, you just solve for x.
When you look at the diagram above, look at the point on the x axis. The X is somewhere, but the Y, will always be ZERO.
So, find the X-intercept of 3x + 4y - 12 = 0 (Remember, Y=0)
3x + 4y - 12 = 0
3x + 4(0) - 12 = 0
3x + 0 - 12 = 0
3x - 12 +12 = 0 +12
3x/3 = 12/3
x = 4 --> (4 , 0)
so now, we want to know what the coordinates of the Y axis is.Now, just like first one, Y is a number, but X will always be ZERO. So, solve for y.
First, the intercepts method.
(By the way, you know what intercept means right? Well, if not, it means "to run into" or "block" )
And thats what the intercepts method means. You look for the two points where the line crosses the X axis, and the Y axis.
Once you find the line, you'll find your coordinates.
A question can look like this = "3x + 4y - 12 = 0"
To find your coordinates, you want to find the value of X, then the value of Y.
To find the coordinates, first you wanna find the X axis. to find it, you just solve for x.
When you look at the diagram above, look at the point on the x axis. The X is somewhere, but the Y, will always be ZERO.
So, find the X-intercept of 3x + 4y - 12 = 0 (Remember, Y=0)
3x + 4y - 12 = 0
3x + 4(0) - 12 = 0
3x + 0 - 12 = 0
3x - 12 +12 = 0 +12
3x/3 = 12/3
x = 4 --> (4 , 0)
so now, we want to know what the coordinates of the Y axis is.Now, just like first one, Y is a number, but X will always be ZERO. So, solve for y.
3x + 4y - 12 = 0
3(0) + 4x - 12 = 0
0 + 4x - 12 = 0
4x - 12 +12 = 0 +12
4x/4 = 12/4
y = 3 --> (0 , 3)
now we can graph it onto our grid.
And that's how it's done
.
does the name look familiar? Its because the formula of the slope intercept method, includes the "slope method" and "the intercept method", and its put into one equation.
Miss Armstrong says: "By looking at the equation, you should know what the graph looks like.
Anyways, the formula is "y = mx + b
y is the y axis
m is the slope
x is the x axis
and b is the y intercept
STEP 1 = get the equation into y = mx + b form
3x + 4y - 12 = 0
3x + 4y -4y -12 = 0 -4y
(3x - 12)/-4 = (-4y) /-4
¾ x + 3 = y
STEP 2 = plot the y unit
STEP 3 = from b, find the second point using the slope method
aaaaand there you go.
homework for today was excercise 11 and 12.
please comment on my blog, i dont know if i made sense or not
next to blog is ... Natnele =)
Labels: angela, intercept method, scribe, slope intercept method
Monday, March 05, 2007
posted by Jenny at 11:54 p.m.
Factor: 2x^2 + 7x - 4
First, times the terms a and c. Second, find factors of ac that add up to b.
In general, remember that "ax^2 + bx + c" means you have to find the factors of ac that add up to b
First, times the terms a and c. Second, find factors of ac that add up to b.
Add the two factors into the equation, then fully factor the equation. You can use the FOIL method to check if your answer brings you back to the original equation.
Don't get it? Don't worry =)
check out this helpful link:
http://www.rock.uwc.edu/facultypages/galexand/baw/thirteen/lesson13.htm
check out this helpful link:
http://www.rock.uwc.edu/facultypages/galexand/baw/thirteen/lesson13.htm
Labels: grouping method, Jenny, scribe
posted by Anonymous at 7:56 p.m.
HI! i'm going to talk about difference of squares. As you can see in the picture, factoring difference of squares is very easy. You just have to know your perfect squares and there's no doubt, you can do difference of square in just a second. =))
a2 - b2
This expression is called DIFFERENCE OF TWO SQUARES.
(a+b) (a-b) = a2 - b2
How about this one? You may remember seeing expressions like these one when you worked with multiplying algebraic expressions.
If you know this fact, then you already know that the factors of
a2 - b2
are
(a - b) (a +b)
REMEMBER: An algebraic term is a perfect square when the numerical coefficient (the number in front of the variables) is a perfect square and the exponents of each of the variables are even numbers. =)
EXAMPLES:
1. Factor x2 - 9
Both x2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares.
What times itself will give x2 ? The answer is x.
What times itself will give 9 ? The answer is 3.
These answers could also be negative values, but positive values will make our work easier.
The factors are (x + 3) and (x - 3).Answer: (x + 3) (x - 3) or (x - 3) (x + 3)
2. Factor x4 – 1
So the square root of x4 is x2 and the square root of 1 is 1. So you'll end up with (x2-1) (x2+1).
As you can see, we're not done yet. We can still factor x2-1 into (x-1)(x+1).
Now it's completely factored.
So the answer'll be (x-1)(x+1)(x2+1)
So now that I've explained what difference of squares is, maybe you want to try more exercises to master your factoring skills. So that's it. That's how easy factoring of difference of squares is. In my opinion, its the easiest and fastest factoring method. So, enjoy! =)
-janelle :)
(a - b) (a +b)
REMEMBER: An algebraic term is a perfect square when the numerical coefficient (the number in front of the variables) is a perfect square and the exponents of each of the variables are even numbers. =)
EXAMPLES:
1. Factor x2 - 9
Both x2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares.
What times itself will give x2 ? The answer is x.
What times itself will give 9 ? The answer is 3.
These answers could also be negative values, but positive values will make our work easier.
The factors are (x + 3) and (x - 3).Answer: (x + 3) (x - 3) or (x - 3) (x + 3)
2. Factor x4 – 1
So the square root of x4 is x2 and the square root of 1 is 1. So you'll end up with (x2-1) (x2+1).
As you can see, we're not done yet. We can still factor x2-1 into (x-1)(x+1).
Now it's completely factored.
So the answer'll be (x-1)(x+1)(x2+1)
So now that I've explained what difference of squares is, maybe you want to try more exercises to master your factoring skills. So that's it. That's how easy factoring of difference of squares is. In my opinion, its the easiest and fastest factoring method. So, enjoy! =)
-janelle :)
Labels: difference of squares, polynomials and factoring, scribe
posted by Maria at 4:47 p.m.
+copy.jpg)
Click the picture
Factoring Polynomials
Factoring a polynomial is the opposite process of multiplying polynomials. Recall that when we factor a number, we are looking for prime factors that multiply together to give the number; for example
6 = 2 ´ 3 , or 12 = 2 ´ 2 ´ 3.
When we factor a polynomial, we are looking for simpler polynomials that can be multiplied together to give us the polynomial that we started with. You might want to review multiplying polynomials if you are not completely clear on how that works.
§ When we factor a polynomial, we are usually only interested in breaking it down into polynomials that have integer coefficients and constants.
Simplest Case: Removing Common Factors
The simplest type of factoring is when there is a factor common to every term. In that case, you can factor out that common factor. What you are doing is using the distributive law in reverse—you are sort of un-distributing the factor.
Recall that the distributive law says
a(b + c) = ab + ac.
Thinking about it in reverse means that if you see ab + ac, you can write it as a(b + c).
Example: 2x2 + 4x
Notice that each term has a factor of 2x, so we can rewrite it as:
2x2 + 4x = 2x(x + 2)
Difference of Two Squares
If you see something of the form a2 - b2, you should remember the formula
Example: x2 – 4 = (x – 2)(x + 2)
- This only holds for a difference of two squares. There is no way to factor a sum of two squares such as a2 + b2 into factors with real numbers.
Algebra Planet Blaster
posted by Lamael at 3:52 p.m.
FINDING GCF
Greatest Common Factor
- is the largest number that is a common factor of two or more numbers
Greatest Common Factor
- is the largest number that is a common factor of two or more numbers
To find the greatest common factor:
- Determine if there is a common factor of the numbers.
- Divide both of the numbers by this common factor.
* If there are no common prime factors, the GCF is 1.
Example:
12m² + 6mn + 3n²
Step 1:
12m² = 3.2.2.m.m
6mn = 3.2.m.n
3n² = 3.n.n
(All have "3" in common)
GCF : 3
Step 2:
3 ( 4m² + 2mn + n² )
Usually you can find the Greatest Common Factor fairly easily by experimenting with possible divisors:
Start with the smaller number; it is the largest divisor of itself. Is it a divisor of the larger number? If so, you have the G.C.F.; if not: What is the next-largest divisor of the smaller number; is IT a divisor of the other number? Continue until you find a number that will divide into BOTH. Sometimes only the number '1' will work as a common divisor; for example: 21 and 16 have no common factor other than 1.
posted by Xochitl at 1:02 p.m.
hi everyone! :) How are you? I'm here to help you with your math problems! (limited to the midpoint formula). Ok, so when using the midpoint formula it is very important to remember a few things:
*always keep the x1 with the y1 and the x2 with the y2
*your adding the x's and the y's so if you have a negative don't get confused (I use brackets becauase it keeps my mind focused and off of muffins! :P)
*don't add your x and y answers like you do in the distance formula, keep them seperated, remember your trying to get a point, so you should get two numbers
example;
formula for Mpt. = (x1+x2 , y1+y2) formula for distance = (the square root sign) (x2-x1) 2 + (y2-y1) 2
*always keep the x1 with the y1 and the x2 with the y2
*your adding the x's and the y's so if you have a negative don't get confused (I use brackets becauase it keeps my mind focused and off of muffins! :P)
*don't add your x and y answers like you do in the distance formula, keep them seperated, remember your trying to get a point, so you should get two numbers
example;
formula for Mpt. = (x1+x2 , y1+y2) formula for distance = (the square root sign) (x2-x1) 2 + (y2-y1) 2
Labels: analyinic geometry, scribe, xochitl
