Chapter 3: Linear Functions

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Chapter 3 Discussion misterlamb misterlamb 0 80 Oct 16, 2008 by misterlamb misterlamb

Summary Assignments

Section 3-1: Paige H. and Josh H.
Section 3-2: Amira A. and Trisha T.
Section 3-3: Andrew E. and Jacob L.
Section 3-4: Jeremiah B. and Jessica R.
Section 3-5: Abigail B. and Tyler Y.
Section 3-6: Arron H. and Amanda T.
Section 3-7: Nate J. and Nick Y.
Section 3-8: Tricia L. and Jacob R.
Section 3-9: Ryan B. and Isaiah C.

Section 3-1: Constant-Increase of Constant-Decrease Situations

AA Section 3-1
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Section 3-1 was about constant-increase and constant-decrease situations. First of all,
we learned about linear equations, which is an equation that gives the graph of a line.
Then we learned that slope intercept form is an equation in the form of y=mx+b, where
m= the slope and b= the y-coordinate of the y-intercept. In this form, the slope will
always be with the independent variable and the y-coordinate of the y-intercept will always
be by itself. We also learned about linear functions, which are functions in the form of
y=mx+b. In Euler notation, it looks like f(x)= mx+b and in Mapping notation it looks like
f:x--> mx+b. Lastly, we learned about a Piecewise Linear Graph, which is when the rate
of change switches from one constant value to another. It is made of two or more segments or rays.
-Paige H
In this lesson we learned about Constant-Increase and Constant-Decrease. First we learned about Linear Equations;which is a equation that gives a graph of a line. After we learned about the linear equation we learned about the initial condition. The initial condition is where we start out at. Next we learned about the Slope-intercept form. The form is y= mx+b, where m=slope and b= the y coordinate of the y-intercept. We then learned the linear function, which is a function of the form y=mx+b. You can put the equation in two ways the Euler and Mapping Notations. The Euler Notation is f(x)=ms+b and the Mapping Notation is f:x on to mx+b. Finally, we learned about the Piecewise Linear Graph, which is when the rate of change switches from one constant value to another.
-Josh H.

Section 3-2: The Graph of y = mx + b

AA Section 3-2
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In section 3-2, we examined Slope-intercept form a little more. We started with some warm-up problems including the example: 3y = 9x - 12 for which we had to solve for y. 3y ÷ 3 = (9x - 12) ÷ 3 y = 3x - 4
Then, as further practice, we solved some problems where we had to state the slope and y-intercept of the equation by observing it in Slope-intercept form. In Slope-intercept form, y = mx +b, y stands for the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept. Remember that the y-intercept is the point in which the graphed equation crosses the y-axis. When you graph the equations it is also important to remember to label each axis and to use a straight edge when drawing any straight lines. We also practice graphing with our graphing calculator to check our work.
Next, we moved to examining equations that create vertical and horizontal lines when graphed. An equation for a horizontal line is in the form y = c where c is constant, while the equation for a vertical line is in the form x = c where c is constant. Don't forget that the slope for a vertical line is 0 and the slope for a horizontal line is undefined. Then after some practice, we recieved two theorems that stated that if two lines have the same slope then they're parallel, and vice versa. At the end of the lesson, we recieved our homework which was pg. 149 # 1 - 24
-Trisha T.

Section 3-3: Linear-Combination Situations

AA Section 3-3
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Section 3-3

Lesson 3-3 is about linear combination situations. In a linear combination all variables are to the first power and are not multiplied or divided by each other. These situations occur whenever quantities are mixed, such as medicines, or chemicals. An expression of this would be Ax+By and A and B are both constant. Andrew E.

Linear Combination:
~ An expression of the for Ax + By, where A and B are constants.
~ All variables are to the first degree.
~ The variables are not multiplying or dividing each other.
~ Gives the graph of a line (or line segment).
ex. Jack bought $200 worth of fruit at Jiffy-Spiffy-Wiffy-Mart (yeah that's right, $200 worth!) He only bought apples and oranges. Apples are $2 each, and grapefruits are $5 each. (Jiffy-Spiffy-Wiffy-Mart is very expensive!) Write an equation representing this situation using the variables A for apples, G for grapefruits, and T for total cost:
T = 2A + 5G 200 = 2A + 5G
Find five possible outcomes by making a table: (These points from the table can then be graphed by simply plotting them.)
A: 0 10 20 50 100
G: 40 36 32 20 0
ex. There is a drink Kevin wants to make. It envolves two different food-colors, Y yellow and R red. 20% of the amount of yellow is used and 40% of the amount of red is used. (T will represent the total of the combinations of the food-colors.) Use the equation T = .2Y + .4R. Solve for the amount of red color that must be used if Y = 400 and T = 100...
100 = .2 x 400 + .4R (subtract 80 from both sides) 20 = .4R (divide both sides by .4) R = 50

So that's 3-3... Go Math!
~ Jacob L.~

Section 3-4: The Graph of Ax + By = C

AA Section 3-4
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In section 3-4:
~Standard Form of an Equation for a Line:
~Ax+By=C, where A, B does not equal 0 is a line. (Shocking Theorem: The graph of Ax+By=C, where A, B does not equal 0 is a line.)
~Slope-intercept: We can represent oblique and horizontal lines, but not vertical.
~Standard form: We can represent all types of lines.
~Another benefit of standard form: We can graph by intercepts.

Section 3-5: Finding an Equation of a Line

AA Section 3-5
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Section 3-5 is about finding the equation of a line. The things that determine a line are 2 points. Once you know 2 points, you can find an equation for the line. The point-slope theorem says that if a line contains (x1,y1) and has slope m then it has the equation y-y1 - m(x-x1). You need only a point and the slopeof the line to figure out an equation. Instead of using slope-intercept form, it is much faster to use slope-point when you are given just the slope and one point. -Abigail B.

Section 3-6: Fitting a Line to Data

AA Section 3-6
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In this lesson we learned how to fit a line to the data. Also in this lesson we learned the meaning of three words; Scatterplot, Line of Best Fit/Regression Line and Coefficient of Correlation. A scatterplot is simply the collection of data that is scattered that describes a situation. Line of best fit or regression line is a line that comes close to most of the points and the coefficient of correlation determines the strength of the relationship between the variable. We also learned the steps to graph a scatterplot and how to find the regression line.
Amanda T.

Section 3-7 and 3-8: Recursive and Explicit Formulas for Arithmetic Sequences

AA Section 3-7/3-8
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In this section we learned about recursive formulas. A recursive formula has an arithmetic sequence. This is a sequence with a constant difference between terms and the domain of natural numbers. Recursive formulas also have a theorem. This theorem states that a sequence defined by the recursive formula is the arithmetic sequence with first term a1 and constant difference d. Sequences can also be generated by calculator by repeating the steps learned in 3-7. Nate J.
In this lesson we learned about making explicit formulas for arithmetic sequences. Arithmetic sequences have a constant increase that can be represented by multiplying the first term by a specified number of times. Therefore if the constant f increase is 4 and the first term is 543 then the explicit equation is an=543+4(n-1).
J. Rita
For lesson 3-8 we learned how to make an explicit formula for an arithmetic sequence. To do this we learned a Theorem which states: The nth term an of an arithmetic sequence with 1st term a1 and a constant difference d then you have a formula that looks like an=a1+(n-1)d. We also learned that you can move from a explicit formula to a recursive formula and back.
-Tricia L.

Section 3-9: Step Functions

Step Functions
In this lesson we learned about step functions and rounding down functions. The rounding down function deals with integers and the way they are rounded. When the number is a negative number but has a decimal after it the number will go to the next lowest decimal number. If the number is positive and has a decimal the number will just stay whatever the number is before the decimals. When it comes to the step functions there is a function that is defined by f(x) = [x] and in this function you insert the data that is given to you by the problem and figure out where to plot the points on a graph.
-Isaiah C.
-Ryan B.

AA Section 3-9
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