This Blog exists for the collective benefit of all algebra students. While the posts are specific to Mr. Chamberlain's class, any and all "algebra-ticians" are welcome. The more specific your question (including your own attempts to answer it) the better.
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Friday, January 7, 2011
hw #5-5 Parallel, Perpendicular, and Direct Variation
hw #5-5
pg 330-32 #1-4 All
#7-35 Odd
pg 302-03 # 1-5 All
#9-14 All
#30-34 All
The short answer is "Yup" - you are correct. Of course, be careful when blogging... y=(-1/2)x is different than y= -1 over 2x... sometimes blogging and math don't mix well!!
Direct variation can be a difficult concept to grasp. In the end, the question is can you represent the relationship between x and y as an equation in the form y = kx, where k is some real number.
Direct variation is another cousin of y=mx+b, where m and b are "some" real numbers.
Essentially, direct variation will yield a linear equation where the y-intercept (aka 'b') is ALWAYS zero.
Direct variation is sometimes represented as the equation:
a) y = kx ... and other times represented as an equation/proportion
b) y/x = k
... accomplished algebra-ticians such as yourselves will recognize that the difference between equations a) and b) above is simply taking equation a) and dividing both sides by x... i.e. they are not DIFFERENT equations, they can easily be DERIVED from one another... ca-peesh?
The opposite of x is the opposite of x, sometimes called negative x... written as -x, yes? On a number line, you can find the opposite of a number by counting your distance to zero and then continuing past zero for the same distance, yes?
A real number and its opposite have the the same absolute value, i.e. distance from zero, yes?
The reciprocal of x is 1/x, since (x)(1/x)= 1, yes?
Formal Defn: A number multiplied by its reciprocal = 1.
The negative reciprocal of x is 1/-x, since (x)(1/-x) = -1, yes? As we discussed on Friday, the negative reciprocal can also be called the opposite reciprocal.
Formal Defn: A number multiplied by its negative (or opposite) reciprocal = -1.
For #13, I checked the back of the book for the answer, and it said it was perpendicular because -x is the opposite reciprocal of x.. But wouldn't it be 1/-x?
You are making a common mistake for neophyte algebra-ticians.
y = mx + b, agree?
Slope is the 'm' NOT the 'mx'
So, the linear equations y=x+2 and y=-x+4 graph as perpendicular lines because the SLOPES (1 and -1 respectively) are negative reciprocals. (see definitions above)
You were looking at mx, instead just m... ca-peesh??
It's a common error, but we have to lock it into our Algebra Toolkit sooner than later.
As we discussed on Friday, horizontal (y= some#) and vertical (x= some#) lines, while clearly perpendicular, are the ONE exception to the negative reciprocal rule. Can you explain why these lines violate this rule?
Horizontal and vertical lines do not have slopes that are opposite reciprocals BECAUSE the slope of a vertical line is UNDEFINED... IT IS NOT A NUMBER!!!
Don't say I didn't warn you that horizontal lines and vertical lines will give you the MOST TROUBLE!
From an algebraic perspective, this is simply a formula (aka literal equation, right?),
V = IR
a) is asking you to solve for Volts when I=24 and R=2... can you multiply 24 and 2 to get the number of Volts?
b) is asking you to create an equation for R in terms of V and I and then substitute 24 for I and 18 for V... can you solve the Volts formula for R= and perform the required substitution?
A direct variation means there is a CONSTANT RATIO between the two variables. CONSTANT means that the ratio does NOT change, even as the value of the variables changes.
For example, if I got paid $12 each time one of my students got an A on a quiz, I would get $24 for 2 A's, $144 for 12 A's, and $1212 for 101 A's. In this example the ratio of dollars to A's is 12/1... it never changes no matter how many A's my students earn.
Algebraically, at this rate, I will never be able to afford retirement.
The ratio of 160/4.6 will create a repeating decimal... how much precision does this problem really require? Round to two decimal places and see what you come up with.
I am confused on #4 on page 302 i can find an equation im just not sure if its a direct relation or not.
ReplyDeleteI think I got it is the equation y=-1/2x? that would be a direct relation I think.
ReplyDeleteConfusion is good.
ReplyDeleteThe short answer is "Yup" - you are correct. Of course, be careful when blogging...
y=(-1/2)x is different than y= -1 over 2x...
sometimes blogging and math don't mix well!!
Direct variation can be a difficult concept to grasp. In the end, the question is can you represent the relationship between x and y as an equation in the form y = kx, where k is some real number.
Direct variation is another cousin of y=mx+b, where m and b are "some" real numbers.
Essentially, direct variation will yield a linear equation where the y-intercept (aka 'b') is ALWAYS zero.
Direct variation is sometimes represented as the equation:
a) y = kx
... and other times represented as an equation/proportion
b) y/x = k
... accomplished algebra-ticians such as yourselves will recognize that the difference between equations a) and b) above is simply taking equation a) and dividing both sides by x... i.e. they are not DIFFERENT equations, they can easily be DERIVED from one another... ca-peesh?
Note to ALL; If Julia can ask a question, it is clearly OK for ANYONE to have a question...
ReplyDeleteCA-PEEESHHHHHH????!!!!
The BIGGEST PROBLEM we have as a class is the LACK OF QUESTIONS.
Ca-peesh :)
ReplyDeleteo....wow lotta
ReplyDeleteWould the opposite of x be -1/x?
ReplyDeleteJustin- Well, I'm pretty much usually the only one asking questions.. So I got it. :)
ReplyDeleteThe opposite of x is the opposite of x, sometimes called negative x... written as -x, yes? On a number line, you can find the opposite of a number by counting your distance to zero and then continuing past zero for the same distance, yes?
ReplyDeleteA real number and its opposite have the the same absolute value, i.e. distance from zero, yes?
The reciprocal of x is 1/x, since (x)(1/x)= 1, yes?
Formal Defn: A number multiplied by its reciprocal = 1.
The negative reciprocal of x is 1/-x, since (x)(1/-x) = -1, yes? As we discussed on Friday, the negative reciprocal can also be called the opposite reciprocal.
Formal Defn: A number multiplied by its negative (or opposite) reciprocal = -1.
Ca-peeesh?
Word to the Wise: If you don't understand point-slope form, you better get your butt in gear and work some problems.
ReplyDeleteFrom,
Your Favorite Quiz Grader
Hint: When do you use point-slope form? When you're given a point and a slope!
Like (-33, -8) and m = 1/3
Sound familiar?????
Yup, that's what I meant, opposite reciprocal.
ReplyDeleteThank you!
I know point-slope form pretty well.. And the quiz was pretty easy.. So I'm crossing my fingers!
For #13, I checked the back of the book for the answer, and it said it was perpendicular because
ReplyDelete-x is the opposite reciprocal of x.. But wouldn't it be 1/-x?
For #17, I'm not sure how to explain why it is "neither".
ReplyDeleteDear #13,
ReplyDeleteYou are making a common mistake for neophyte algebra-ticians.
y = mx + b, agree?
Slope is the 'm' NOT the 'mx'
So, the linear equations y=x+2 and y=-x+4 graph as perpendicular lines because the SLOPES (1 and -1 respectively) are negative reciprocals.
(see definitions above)
You were looking at mx, instead just m... ca-peesh??
It's a common error, but we have to lock it into our Algebra Toolkit sooner than later.
Dear #17,
ReplyDeleteThat's because 'neither' is the wrong answer!
As we discussed on Friday, horizontal (y= some#) and vertical (x= some#) lines, while clearly perpendicular, are the ONE exception to the negative reciprocal rule. Can you explain why these lines violate this rule?
Because.. With only one coordinate, one is horizontal & one is vertical, meaning they have to intersect at some point?
ReplyDeleteHopefully?
I don't understand how to write equations for #35.
ReplyDeleteWould #2 on pg 302 be 1/10y=x?
ReplyDeleteDear #17,
ReplyDeleteHorizontal and vertical lines do not have slopes that are opposite reciprocals BECAUSE the slope of a vertical line is UNDEFINED... IT IS NOT A NUMBER!!!
Don't say I didn't warn you that horizontal lines and vertical lines will give you the MOST TROUBLE!
Wait nevermind.. y=10x??
ReplyDeleteOHHHHHHH...
ReplyDeleteI gotsk it.
Not Sure how to write an equation for #32 on pg 303.
ReplyDeleteSincerely,
Not Sure how to write an equation for #2 on pg 303.
Dear #35,
ReplyDeleteFrom an algebraic perspective, this is simply a formula (aka literal equation, right?),
V = IR
a) is asking you to solve for Volts when I=24 and R=2... can you multiply 24 and 2 to get the number of Volts?
b) is asking you to create an equation for R in terms of V and I and then substitute 24 for I and 18 for V... can you solve the Volts formula for R= and perform the required substitution?
I think you can!!
Ca-peeesh?
Dear #2,
ReplyDeleteWell... almost, kinda-sorta... Direct variation equations are typically stated as y=kx or y/x=k... can you solve for y= instead of x=?
lmk
I think you're talking about a different #35..
ReplyDeletePage 332 #35? Sorry.
And for #2.. y=10x?
Trouble with #34.. Page 303.
ReplyDeleteDear #34,
ReplyDeleteA direct variation means there is a CONSTANT RATIO between the two variables. CONSTANT means that the ratio does NOT change, even as the value of the variables changes.
For example, if I got paid $12 each time one of my students got an A on a quiz, I would get $24 for 2 A's, $144 for 12 A's, and $1212 for 101 A's. In this example the ratio of dollars to A's is 12/1... it never changes no matter how many A's my students earn.
Algebraically, at this rate, I will never be able to afford retirement.
lmk if this helps...
I can't seem to find a constant ratio between 160 & 4... Only a LOOOOOONG decimal.
ReplyDeleteDear #34,
ReplyDeleteThe ratio of 160/4.6 will create a repeating decimal... how much precision does this problem really require? Round to two decimal places and see what you come up with.
Does the word 'proportion' ring a bell here??
160 is to 4.6 as ? is to ?
Ca-peesh?
Ca-Peesh.
ReplyDelete