Chapter 3: Linear Functions



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Chapter 3 Discussion misterlamb misterlamb 0 89 Oct 16, 2008 by misterlamb misterlamb

Summary Assignments

Section 3-1: Crystalene E. and Elissa M.
Section 3-2: Kaitlynn R. and Max S.
Section 3-3: Austin R. and Thomas S.
Section 3-4: Kim K. and Dale N.
Section 3-5: Nate F. and Sarah K.
Section 3-6: Justin F. and Marisa M.
Section 3-7: Dominic F. and Caiti T.
Section 3-8: Nick H. and Austin L.
Section 3-9: Mitch K. and Shannon S.

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

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Elissa M
We learned about linear equations. A linear equation is an equation that gives us the graph of a line. We also learned about slope-intercept form. Its equation is y= mx+b where m is the slope and b is the y coordinate of the y intercept. Slope is always the independent variable.
Linear function is a function of the form y=mx+b
Piecewise linear graph is when the rate of change switches from one constant to another and is made up of two or more segments or rays.

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

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y=mx+b is considered the slope intercept form. In this form the m represents the slope and b represents the y intercept. When askeProxy-Connection: keep-alive Cache-Control: max-age=0 20for the y intercept it must be written as a poProxy-Connection: keep-alive Cache-Control: max-age=0 oxy-Connection: keep-alive Cache-Control: max-age=0 oxy-Connection: keep-alive Cache-Control: max-age=0 oxy-Connection: keep-alive Cache-Control: max-age=0 oxy-Connection: keep-alive Cache-Control: max-age=0 oxy-Connection: keep-alive Cache-Control: max-age=0 t. For example: y=2x+4 The slope is 2 and the y intercept in (0,4).
The equation for a horizontal line is y=c, where c is the constant. The equation for a vertical line is x=c, where c is the constant. A theorem that we learned is if two lines have the same slope then the lines are parallel. We also learned a theorem that states that if two lines are parallel then they have the same slope. Kaitlynn R.

In this lesson we learned about the graph of the equation y=mx+b. We also learned that the equation for a horizontal line is y=c where c is a constant, and that the equation for a vertical line is x=c where c is a constant. In addition to this, we learned that if two different lines have the same slope, then they are parallel. If two lines are parallel, then they have the same slope.
Max J. S.

Section 3-3: Linear-Combination Situations

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Thomas S.: This chapter has to do with Linear - Combination as so the title says. It deals with the expresson of the form Ax + By when A and B are constants. A and B arn't multilying or dividing by eachother. This express gives the graph of a line or a segment. When you deal with this expression you usually plug in for x and y to get the output.
Austin R.: Now that Thomas explained how it works I'll try to give a walk through probelm.
consider 3x+4y=24
I usually start with a table to find x and y values when I plug in a number
1 _5 1/4
2 _4 1/2
3 _3 3/4
0 _6
-1 _6 3/4
-2 _7 1/2
-3 _8 1/4
I do not have a graphing progarm so I cannot really give you a graph, but the graph would be very self explanitory, just plug in the cordinates and your done

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

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Dale N.-
Standard Form of a Line-
Ax+By=C, where A, B are not equal to 0.

Slope intercept form can represent oblique and horizontal lines, not vertical.

Standard form represents oblique, horizontal, and vertical.

When we use standard form, we can graph our line by the x and y intercepts. We can do this because you only need 2 points to make up a line

Chapter 3-4 was on graphing Ax+By=C. This form is called standard form. There was also the Shocking theroem. This is the graph of Ax+By=C where A and B and not equal to C. Standard form is also good because you can represent all types of lines. All you have to do is plug either and x or y value in soulve and then do for the opposite and the graph the two points and boda bing u have a line :). Like the ecuation 10x+6y=C if u pig 0 in for x u get 5 and 0 in for y you get 3 so you have the point 0,5 3,0. the graph and you have land. And that is lesson 3-4

Section 3-5: Finding an Equation of a Line

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Alright everyone! Chapter 3 lesson 5. This lesson is all about finding the equation of a line! There is one theorem in this lesson and that is the point- slope theorem. It states that is a line contains a point (X1, Y1) and has slope m, then it has equation Y-Y1=m (X-X1). So let's say your given two points, like (3,5) and (4,8). What you have to do first is to find the slope, which is 8-5/4-3, and will end up being 3. You, then, plug in the slope for m, and pick one point to plug in for Y1 and X1. You then solve for Y and will then have your slope.
- Sarah K.

Chapter 5, lesson 3 involves finding an equation of a line. To do that, you must first know the point-slope theorem. The point-slope theorem is if a line contains (x1, and y1) and has a slope m, then it has equation y-y1=m(x-x1). For example, you have to find an equation for a line that passes through the points (3,5) and (6,-1). You must first find the slope, y1-y2/x1-x2, and if done correctly, you should get -2. Then use the point slope theorem with one of the set of points. In this case I will use (3,5). The equation should look like this, y-5+-2(x-3). From this equation, you can turn it into standard form, or slope-intercept form. Standard form should look like this, 2x=y+11.
Nate F.

Section 3-6: Fitting a Line to Data

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Justin F.
In this lesson we learn how to fit a line to data. A scatterplot is a collection of discrete points that describe a situation. The line you fit to the scatterplot is a regression line. It is a line that comes close to most of the points in the scatterplot. This line does not need to touch any of the points. The coefficient of correlation determines the strength of the relationship between the variables. This will be on -1 is less than or equal to r which is less than or equal to 1. If r<0, the slope is negative and when r>0, the slope is positive. With the equation of the line you can estimate what something will be.

3-6 summary by Marisa M.

Scatterplot: a colletion of discreate points that describe a situation.
Line of best fit/ regression line: A line that comes close to most of the points on the scatterplots. Doesn't need to touch any points.
Coefficient of correlation: Determines the strength of the relationship between variables. Will be on -1< or equal to t < or equal to 1 when r<0 the slope is negative when r>0 the slope is positive.
Follow these steps to graph the data for fitting a line.
First you will enter the values for time in L1 and bpm in L2. Make sure that you enter the values in the same order for both lists. Now press 2nd then stat plot to graph this information. Got to Plot 1 and press enter. Turn this plot on and select the scatter plot shown without having to worry about setting up a window. We can check the correlation by making a line of best fit. First we need to turn on the diagnostic. Press 2nd then catalog and find diagnosticon and press enter. Now press stat and go to teh calc menu. Find Linereg(ax+b) and presss enter. We need it to find the line of best fit for the list of L1 and L2 so we need to enter those on our screen as well, placing commas after each list. we also need to have the equation so we press vars and go to the y-vars menu and find function...we want y1. Now your screen should have all of these then press enter. you will se ethe main screen what y,a,b are equal to. When you press y= you will see an equation for the line of best fit.

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

Lesson 3-7 is mainly about recursive formulas. That is a formula that gives a starting point and an equation to get any term after that first one using the previous term. This was more of a review for us so i wont explain it a whole lot more. The newer vocab words we were given were those such as Arithmetic Sequence which is a sequence with a constant difference between terms, the domain being all natural numbers. A second term is Theorem which is a sequence defined by the recursive formula: a to the nth term -1+d, for int. n> 2. It gives the first term and the constant difference. As was previously said, this section was more of a review or a use of old ways.
Caiti T.
Austin L.

Lesson 3-7 tells us about recursive formulas. Basically with the recusive formula you get a starting point in the form of a term. After you have one term you can keep finding the next terms by using the previous.
THEROEM- a to the nth term -1+d, for int. n> 2. That gives a starting term to work with and the constant difference.
Arithmetic Sequence- it is basically a sequence with a constant difference.
We learned alot of this in our early years so not much more needs to be said.
Dominic F.

Lesson 3-8 is all about Explicit Formulas for Arithmetic Sentences. This might sound like a complicated thing but it really isn't as long as you know the definition which is: the nth term of an arithmetic sequence with 1st term a and constant difference d, is given by the explicit formula. In a nutshell this basically is stating an explicit formula for arithmetic sequences allow you to solve for a specific term without having to solve for all the terms in between. The equation is the nth term of a=the 1st term of a+9(n-1)d.

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3-8 Explicit Formulas for Arithmetic Sequences- Nicholas Asa H.
The lesson I'm covering deals with explicit formulas for arithmetic Sequences. In my own view it seems to me that the explicit sequnces have carried over since last chapter but instead this time you are going to be using addition dealing with number of terms. This type of formula differs from recursive formulas because instead of having to solve multiple terms you can go right to the term you are wanted to solve and find the answer. It is easier, but it is also harder to create the equations. The theorem stated in this lesson is the one for (nth Term of an Arithmetic Sequence) it states.... The nth term of a An of an arithmetic sequence wih first term a1 and constant difference d is given by the explicit formula.

Section 3-9: Step Functions

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Shannon S. pd 3: Step functions are graphs that look like steps and are based off of functions with the equation y is equal to the greatest-integer less than or equal to x. This graph must and will pass the vertical line test and each step has only one endpoint. The greatest-integer function is where f of x is equal to the greatest-integer of x. It can be used in real life when you have an item that can't be rounded up such as t-shirts and pennies. To start graphing the equation, you must pick two numbers and plug each into the given equation. This will tell you which endpoint will be included in the function. To check this, you can plug in any number between the two that you picked. To graph, you will place an open point at one end and a solid point at the end that you found to be included in the function. Draw a horizontal line between them and proceed to plot them this way. For more practice, this lesson is in the book on page 186.

Mitch K. pd 3: more for graphing. keep plugging in two integers until you found a pattern down the right side of the table, or with the outputs. There should be a constant of difference and that should be the space between each step vertically. To find the point included and the point with the open circle at the end, pick a point in between the two integers to see which one is included.