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toc =Chapter 2: Variation and Graphs=





Discussion
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Section 2-1: Direct Variation
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Section 2-1 was about Direct Variation. Direct variation function is a function with a formula of the form y=KX^n, with K not equal to 0, and n>0. In this lesson we learned that r varies directly as c in the equation of r=10c. Also k stands for the constant of variation. A Direct Variation Function can also be know as "Direcly Proportional". -Jon K.

Section 2-1 was about direct variation which is a function with a formula of the form y=kx^n, with k can't equal 0 and n >0. We also learned that k is a nonzero constant, called the constant of variation, and n is a positive number. We also learned that when y varies directly as x^n we say that y is directly proportional to x^n. There are four steps to figure out the equation, first write an equation that describes the variation. Second find the constant of variation, third rewrite the variation function using the constant of variation. Last evaluate the function for the desired value of the independent variable. Andrew E.

Section 2-2: Inverse Variation
media type="custom" key="2054750" In lesson 2-2 we learned that an inverse variation function is a function with a formula pf the form y equals k over x to the n^th with k ≠ 0 and n > 0. We learned that k is always the unknown and we use a previous equation to find k. An Example is d=k/w so 2=k/55. k=110. Then the formula can be used to find any other term that deals with the same problem. -Isaiah C.

Lesson 2-2 was about inverse variation. An inverse variation is a function of the form y=K over x to the n^th with K≠o and n>0. K is always the unknown and you can use a previous equation to find k. When you find k, you can find the other variables in the equation. -Tyler Y.

Section 2-3: The Fundamental Theorem of Variation
media type="custom" key="2059354" 2.3 - You can use the Fundamental Theorem of Variation to find the independent or dependent variables. If one variable is doubled then all of the other variables in the set also have to be doubled. - Courtney G.

-Emma R.?

Section 2-4: The Graph of //y = kx//
In this lesson we learned how to graph y=kx. The slope is the steepness of a line. You can find the slope by Rise over Run which equals the change in y over the change in x. Example m=y1 minus y2 over x1 minus x2. The theorem of the graph of a direct variation equation y=kx is a line with slope k and contains the orgin (0,0). k>0- uphill L to R. k<0= downhill L to R. Here are some examples of how to find a slope. You have (5,-2), (3,8) so you rise over run. You 8-(-2) over 3-5 which equals 10 over -2. Then the slope would be -5. Josh H. and Chris S.

Section 2-5: The Graph of //y = kx²//


-Jeremiah B. and Tricia L.?

Section 2-6: The Graphs of //y = k/x// and //y = k/x//²
In this lesson we learned to graph the equations y=k/x and y=k/x². When you graph the equation y=k/x you get a hyperbola. When you graph the equation y=k/x² you get an inverse-square curve graph. In both of these graphs there are asymptotes, which are lines on a graph that come close, but do not touch. We also learned that the two curves of the inverse variation graphs are called branches. Nick Y.

Section 2-6 was about the graphs of y=k/x and y=k/x². The graph of y=k/x will always be a hyperbola. When k>0, the points will be graphed in the I and III quadrants. When k<0, the points will be graphed in the II and IIII quadrants. The hyperbola will never touch either axis, and will never go through the origin. The two curves that make up an inverse variation graph are called branches. A discrete graph is a graph that is made up of disconnected dots. The graph of y=k/x² will always be an inverse-square curve. When k>0, the points will be graphed in the I and II quadrants. When k<0, the points will be graphed in the III and IIII quadrants. Just like a hyperbola, an inverse-square curve will never touch either axis or go through the origin. An asymptote is a line on a graph which the line comes close to, but never touches. -Abigail B.

Section 2-7 and 2-8: Fitting a Model to Data
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Answers for homework from 2-7/2-8

2/7 and 2/8 Notes

Mathematical Model is a mathematical representation of a real-world situation. In this lesson, it is wise to graph he problem. Once the graph is complete, you can then decide on what type of graph it is (line, parabola, hyperbola, or an inverse-square curve). Then you can analyze the graph and decide on what the formula is for that specific graph. Then find K and plug it back into the formula. Use another point on your graph to see if your formula is correct. Then you can finally state the model of the problem.

CONVERSE OF THE FUNDAMENTAL THEOREM 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, then y varies directly as the nth power of x, or y = k times x to the nth power.

b. If multiplying every x-value of a function by c results in dividing the corresponding y-value by c to the nth power, then y varies inversely as the nth power of x, or y = k divided by x to the nth power.

~ Jake L ~

- Trisha T.?

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! media type="custom" key="2106151"

Lesson 2-9 was about combined and joint variations. A combined variation is when a direct and inverse variation both occur in a problem. It is in the form of y=kx/z, which means that y varies directly as x and inversely as z. To find k in a combined variation problem, you keep all but one of the independent variables constant. Then choose a value for the other independent variables and plug-in. A joint variation is when the dependent variable varies directly as two or more independent variables, but there is no inverse variation. It is in the form of y=kxz³, which means that y varies directly as x and the cube of z. To find k in a joint variation problem, you use the same procedure as in a combined variation problem.

-Paige H.

- Jake R.?

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