Chapter 2: Variation and Graphs


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Chapter 2 Discussion misterlamb misterlamb 0 137 Oct 16, 2008 by misterlamb misterlamb


Section 2-1: Direct Variation


AA Section 2-1
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Mitch K.
Steps to solving a direct variation problem:
1: write equation to describe the variation
2: find k
3: rewrite function using k
4: evaluate
y=kx
y=dependent variable
x=independent variable
k=constant

Sarah P.
: when r gets larger, so does c. when r gets smaller, so does c.
-The constant variation is when k is a nonzero constant in y=kx to the nth power; n is a positive integer.
diret variation function : a function of the form y=kx to the nth power with k not equal to zero and n greater than zero.
-also known as "directly proportional"
EX: the cost of gas varies directly as the amount of gas purchased
-->
the more you get, the more it costs.

Section 2-2: Inverse Variation


AA Section 2-2
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Thomas S.: This lesson was about inverse variation the fomula is y = k/x to the n power. Solving inverse variation is
very similar to solving Direct variation except instead of using multiplication you use division. You solve for "y'" in
this equation
my dividing k by x to the n'th power. When a number is imputed into the equation a number will increase while another
number decreases.

Dale N. - Inverse variation was the talk of this lesson. So, we learned the definition of the inverse-variation function. The inverse-variation function is a function with a formula of the form y=k/x to the nth power, with k not equal to 0 and n>0. Also, we learned how to solve inverse-variation problems. We learned that the law of the lever formula is d=k/w. In order to solve for d, you need to find k first. After you find k, you substitute your number and solve for d. D shows the distance you are away from the fulcrum.



Section 2-3: The Fundamental Theorem of Variation


AA Section 2-3
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Sarah K. and Shannon M.!
The Fundamental Theorem of Variation
A. If y varies directly as x to the nth power,(That is, y=kx to the nth) and x is multiplied by c, then y is multiplied by c to the nth power.
Basically, if you have a number, like 5 to the 2nd power, and you multiply it by 3, then y would be multiplied by 3 to the 2nd power or 9.
B. If y varies inversely as x to the nth power (that is, y=k divided byx to the nth power.) and x is multiplied by a non zero constant c, theny is divided c to the nth power.
So, if you have a number, such as 4 to the 2nd power and it is multiplied by 6, then y would be divided by 6 to the 2nd power or 36.

Section 2-4: The Graph of y = kx


Characteristics of a graph:
-If a graph starts from the bottom left & goes to the upper right, then the graph is positive
-If a graph starts from the top left & goes to the bottom right, then the graph is negative
-May intersect at the origin
Slope-Steepness of a line

To figure out the slope you take the rise over run, or rise divided by run
Another way of wording this could be:

Change in y over change in x
In other words...

y1-y2 over x1-x2

The graph of a direct variation equation y=kx is a line, with slope k, contains the origin
K>0 Uphill left to right
K<0 Downhill left to right
-Tarah L.

Properties of y=kx
The domain and range of the function y=kx is all real numbers when k doesn't equal zero because the lines will continue in the direction it is going. When k equals zero the domain is all real numbers and the range is zero. The origin of the line will always be (0,0) as well. When k is greater then zero it will always go through quadrants one and three and when k is less than zero it will always go through quadrants two and four.

Slope
The slope of a line is the steepness for example in a hill or mountain. The slope is the change in dependent variable over the change of the independent variable. It can be written in many ways like Rise over run or the change in the vertical distance over the change in the horizontal distance.

Slope Equation
The equation for slope is y2-y1 over x2-x1. In other words, y2-y1 is the change in vertical distance between two y coordinates on the same slope, and x2-x1 is the change of two horizontal distances between two x coordinates on the same slope.

Theorem of y=kx
In a function with a constant slope you can any two points on the line on graph and they will have the same slope.

Section 2-5: The Graph of y = kx²


The whole lesson was essentially about graphing the equation y=kx^2 and rate of change. The rate of change is the slope of the graph. The graph y=kx^2 is a parabola. The parabola must go through the point (0,0) and its line of symmetry is the y-axis. A parabola can open up or down. If k is greater than 0 it opens up and if k is less than 0 it opens down. The domain of the equation y=kx^2 is the set of all real numbers. The range when k > 0 is the set of nonnegative real numbers while the range when k < 0 is the set of nonpositive real numbers. Here is an example of a parabola. - Austin R. and Justin F.

Section 2-6: The Graphs of y = k/x and y = k/x²



Sarah P. & Dominic F.
The graph of every function with an equation of the form y=k/x, where k does not equal zero, is a hyperbola.

branches - two seperate parts that consist in hyperbolas.
-when k > 0, the branches of y=k/x lie in the 1st & 2nd quadrants
-when k < 0, the branches lie in the 3rd & 4th quadrants
(the domain & range are the set of all nonzero real numbers)

discrete - when the points on a graph are not connected.
-the domain is the set of natural numbers to do so.

Inverse-square curve - the graph of y=k/x(squared)
- the domain of every inverse curve is {x:x does not equal 0}
- the range depends on the value of k

Asymptotes
when x = 0, y=k/x and y=k/x(squared) are undefined.
-neither curve crosses the y-axis.
- when x is near 0, the functions are undefined.

Section 2-7 and 2-8: Fitting a Model to Data


AA Notes 2-7 and 2-8
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Crystalene E. and Kaitlynn R.


The mathematical model is a representation of a real world situation and a good model holds true for all the given information.

Steps to finding a model from data: 1) graph the data 2) describe the graph 3) state the equation 4) check other points 6) state the model

Converse of fundamental thrm. of variation :
a) if multiplying every x value of a function by c results in multiplying the corresponding y value by c to the nth power the y varies directly as the nth power of x, y=kx to the nth power
b) if multiplying every x value of a function by c results in dividing the corresponding y value be c to the nth power the y varies inversely as the nth power of x, y=k/h 2

Answers for homework from 2-7/2-8


Section 2-9: Combined and Joint Variation


Begin by reading through Section 2-9 in the book. Then come back here and check out the notes!
AA Section 2-9
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Combined variation--> a situation in which direct and inverse variations occur together.
example: y varies directly as x and inversely as z
y=kx/z
Joint variation--> a situation in which one quantity varies directly as the product of two or more independent variables, but not inversely as any variable.
example: y varies jointly as x and z, or, y varies directly as the product of x and z
y=kxz

~marissa b. && gage a.~

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