We learned about Point-Slope Form of Linear Equations in Algebra 1. Point-Slope Form is Y-Y1= m(X-X1). (X1, Y1) is a point on the line and m is the slope.
If you are given a point and the slope you can plug the numbers into the equation.
For example-
Point-(1,2) Slope- 3
If you plug the numbers into the equation it looks like this-
y-2= 3(x-1)
You are often asked to turn this equation into Slope\Intercept Form. You just solve it like an equation. This is how you would do it with the equation above:
y-2= 3(x-1)
y-2= 3x-3 [Distribute.]
+2 +2 [Add 2 to both sides.]
y= 3x-1
You also are often asked to graph from the equations well.
From the equation y-2= 3(x-1) you can pick out that the point in the equation is (1, 2 ) and the slope is 3. You start at the point (1, 2 ) and go up 3 and over 1 (rise over run) to the next point.
:) :)
Tuesday, January 19, 2010
Monday, January 11, 2010
1/11/010
Today in class, we learned about slope-intercept. Slope-intercept uses the formula:
y=mx+b
where:
m=slope
b=y-intercept.
For example, in the equation y=4x+6:
m=4 and b=6.
y=mx+b
y=4x+6
If you are given a slope: .5 and a y-intercept: 7,
you would use that to write the equation y=.5x+7.
Say you are given 2 points on a graph: (0,3) and (5,-2). You use y1-y2/x1-x2 for the slope: -2-3/5-0=-5/5. So, the slope is -1. You know the y-intercept is 3 because of the point (0,3) so the equation is y=-1x+3.
To graph an equation, y=4x+2, for example, start with x being 0. the y-intercept is 2, so your first point is (0,2). the slope is 4, so either use 4/1 for graphing other points or substitute x for whatever number and solve for y to graph the other (x,y)s.
y=mx+b
where:
m=slope
b=y-intercept.
For example, in the equation y=4x+6:
m=4 and b=6.
y=mx+b
y=4x+6
If you are given a slope: .5 and a y-intercept: 7,
you would use that to write the equation y=.5x+7.
Say you are given 2 points on a graph: (0,3) and (5,-2). You use y1-y2/x1-x2 for the slope: -2-3/5-0=-5/5. So, the slope is -1. You know the y-intercept is 3 because of the point (0,3) so the equation is y=-1x+3.
To graph an equation, y=4x+2, for example, start with x being 0. the y-intercept is 2, so your first point is (0,2). the slope is 4, so either use 4/1 for graphing other points or substitute x for whatever number and solve for y to graph the other (x,y)s.
Friday, January 8, 2010
Daily Scribe
Today in class, we reviewed the concept of slope (rise over run). In addition to this, we learned a formula for finding the slope of a line when the coordinates of two points in the line are given. This is y2-y1/x2-x1.
Example: Find the slope of a line that passes through A(-2, 1) and B(5, 7). To solve this problem, you would call point A point 1, and point B point 2. Then you would substitute the variables for these numbers. We know x is the first number and y is the second number in each coordinate, so you know that x1 is -2 and x2 is 5. We also know that y1 is 1 and y2 is 7. Substitute these numbers for the variables in the formula, and you get 7 -1/5 -(-2) = 6/7. This is the slope. This formula would also work if you assigned the A-coordinates to 2 and the B-coordinates to 1. This way, the formula would be 1-7/-2-5=-6/-7, which is equal to 6/7.
Be cautious that you do not accidentally change the formula to y2-y1/x1-x2. This would mess up the whole thing. Also, be sure not to do x/y instead of y/x. If you have to find the slope of a line that passes through (2, 1) and (2, -4), the answer to this is undefined, because you will get the answer of -3/0. If you ever get an answer divided by 0, the answer is undefined, because you cannot divide anything by 0. A vertical line is undefined.
Example: Find the slope of a line that passes through A(-2, 1) and B(5, 7). To solve this problem, you would call point A point 1, and point B point 2. Then you would substitute the variables for these numbers. We know x is the first number and y is the second number in each coordinate, so you know that x1 is -2 and x2 is 5. We also know that y1 is 1 and y2 is 7. Substitute these numbers for the variables in the formula, and you get 7 -1/5 -(-2) = 6/7. This is the slope. This formula would also work if you assigned the A-coordinates to 2 and the B-coordinates to 1. This way, the formula would be 1-7/-2-5=-6/-7, which is equal to 6/7.
Be cautious that you do not accidentally change the formula to y2-y1/x1-x2. This would mess up the whole thing. Also, be sure not to do x/y instead of y/x. If you have to find the slope of a line that passes through (2, 1) and (2, -4), the answer to this is undefined, because you will get the answer of -3/0. If you ever get an answer divided by 0, the answer is undefined, because you cannot divide anything by 0. A vertical line is undefined.
Wednesday, January 6, 2010
Arithmatic Sequences
Today in class we went over functions, and learned about arithmetic sequences and inductive reasoning.
A function is a relation between a number in the range(the f(x) or y) and the domain(the x). You use the numbers to plot points on a graph. For example:
If you had the function f(x)=2.5x and -8, 5, and 0 as x values, you would substitute in the x values to get the y values.
f(x)=2.5(-8) f(x)=2.5(5) f(x)=2.5(o)
f(x)=-20 f(x)=12.5 f(x)=0
This would mean that the points (-8, -20) (5, 12.5) and (o, o) would be graphed as the function f(x)=2.5x.
Also today learned about inductive reasoning. It is when you recognize a pattern, and can continue it. One example is the numbers 1, 4, 16, 64. You recognize the pattern is times 4, and can continue it(256, 1024).
Last we learned about sequences. A sequence is a number pattern. It is made up of terms. An arithmetic sequence is any sequence where a fixed amount is added to the term before it. Example:
19, 23, 27, 31, 34
The fixed amount added is 4.
Even though it is called an arithmetic sequence, the numbers can go up OR down, as you can add negative numbers.
In the sequence below, the number added, or the common difference, is -8.
42, 34, 26, 18, 10
A function is a relation between a number in the range(the f(x) or y) and the domain(the x). You use the numbers to plot points on a graph. For example:
If you had the function f(x)=2.5x and -8, 5, and 0 as x values, you would substitute in the x values to get the y values.
f(x)=2.5(-8) f(x)=2.5(5) f(x)=2.5(o)
f(x)=-20 f(x)=12.5 f(x)=0
This would mean that the points (-8, -20) (5, 12.5) and (o, o) would be graphed as the function f(x)=2.5x.
Also today learned about inductive reasoning. It is when you recognize a pattern, and can continue it. One example is the numbers 1, 4, 16, 64. You recognize the pattern is times 4, and can continue it(256, 1024).
Last we learned about sequences. A sequence is a number pattern. It is made up of terms. An arithmetic sequence is any sequence where a fixed amount is added to the term before it. Example:
19, 23, 27, 31, 34
The fixed amount added is 4.
Even though it is called an arithmetic sequence, the numbers can go up OR down, as you can add negative numbers.
In the sequence below, the number added, or the common difference, is -8.
42, 34, 26, 18, 10
Tuesday, January 5, 2010
Functions
Today in class we went further into functions. We wer given a function and had to come up with a domane and range for it. To make the function more accurate on a graph you want to make the value of x be both positive and negative numbers. When you come up with your x values you simply put them into ur equation to find what the function of x or f(x) equals.
EXAMPLE: f(x)=4(x)+3
f(-5)=4(-5)+3 x f(x)
-5 -17 f(-5)=-20+3
f(-5)=-17 -1
1
5
Then you would finish all the numbers by doing the same thing and subsituting your x values for x in the equation and your outcome is f(x) or y on the graph. The example done for you would be at the point -5, -17 on a graph.
complete example: f(x)=5x-1 x f(x)
-6 -31
-1 -6
0 -1
1 4
6 29
Also, we were given a domain and a range and asked to find the function. When finding the funcion, alway check your idea on every x, f(x) pair before writing the function for your chart.
EXAMPLES: x f(x) funtion- f(x)=x(4) x f(x)
2 x4 8 2 5
4 x4 16 5 11
5 x4 20 11 23
8 x4 32 23 47
When you are looking for the function it isn't always just one step funtions making them more tricky like the second one. The function is f(x)=2(x)+1.
The other problems we did today were very similar to those in our MSA unit. For these you had to relate, define, and write.
EXAMPLES: Mark recently started a snow shoveling company. He spent $20 on two new shovels and $50 on salt to last the season meaning he put $70 into his business. For each yard he charges $30. Write a fanction to show Mark's profit.
relate- 30 dollars per yard-70 dallars=profit
define- y= yard
p=profit
write- 30(y)-70=p
If mark shovels 5 yards his profit will be 80 dollars. Just like when you find the range of a function you plug your value into the equation.
EXAMPLE: f(x)=4(x)+3
f(-5)=4(-5)+3 x f(x)
-5 -17 f(-5)=-20+3
f(-5)=-17 -1
1
5
Then you would finish all the numbers by doing the same thing and subsituting your x values for x in the equation and your outcome is f(x) or y on the graph. The example done for you would be at the point -5, -17 on a graph.
complete example: f(x)=5x-1 x f(x)
-6 -31
-1 -6
0 -1
1 4
6 29
Also, we were given a domain and a range and asked to find the function. When finding the funcion, alway check your idea on every x, f(x) pair before writing the function for your chart.
EXAMPLES: x f(x) funtion- f(x)=x(4) x f(x)
2 x4 8 2 5
4 x4 16 5 11
5 x4 20 11 23
8 x4 32 23 47
When you are looking for the function it isn't always just one step funtions making them more tricky like the second one. The function is f(x)=2(x)+1.
The other problems we did today were very similar to those in our MSA unit. For these you had to relate, define, and write.
EXAMPLES: Mark recently started a snow shoveling company. He spent $20 on two new shovels and $50 on salt to last the season meaning he put $70 into his business. For each yard he charges $30. Write a fanction to show Mark's profit.
relate- 30 dollars per yard-70 dallars=profit
define- y= yard
p=profit
write- 30(y)-70=p
If mark shovels 5 yards his profit will be 80 dollars. Just like when you find the range of a function you plug your value into the equation.
Monday, January 4, 2010
Functions
Today, we learned about functions, domains, and ranges.
A function is a relation that assigns exactly one value in the range to each value in the domain. To figure out what a domain and range are, we think of the domain as "x" in an ordered pair and range as "y".
When listing domains and ranges, you insert them into brackets. ({})
For example, suppose a graph has 2 columns, one labeled x, and the other labeled y. In the x column, we have the numbers 1, 2, 3, 4, and 5. In the y, we have 2, 4, 6, 8, and 10.
The question is, what is the domain and range?
The domain would be: {1, 2, 3, 4, 5}, and the range: {2, 4, 6, 8, 10}
When listing domains and functions, you want to list them in order and not repeat any numbers.
A function is only allowed one y for each x. So, if a set of ordered pairs is like this: (2,3), (3, 7), (6, 13), (2, 9), it is not a function because 2 has two y's.
To make this a little easier, there is a method called the "stupid pencil trick" where you graph out the coordinates on a graph, hold a pencil vertically over the graph, and see if any y's are on the same line.
A function is a relation that assigns exactly one value in the range to each value in the domain. To figure out what a domain and range are, we think of the domain as "x" in an ordered pair and range as "y".
When listing domains and ranges, you insert them into brackets. ({})
For example, suppose a graph has 2 columns, one labeled x, and the other labeled y. In the x column, we have the numbers 1, 2, 3, 4, and 5. In the y, we have 2, 4, 6, 8, and 10.
The question is, what is the domain and range?
The domain would be: {1, 2, 3, 4, 5}, and the range: {2, 4, 6, 8, 10}
When listing domains and functions, you want to list them in order and not repeat any numbers.
HOW DO YOU KNOW WHAT IS A FUNCTION AND WHAT ISN'T?
A function is only allowed one y for each x. So, if a set of ordered pairs is like this: (2,3), (3, 7), (6, 13), (2, 9), it is not a function because 2 has two y's.
To make this a little easier, there is a method called the "stupid pencil trick" where you graph out the coordinates on a graph, hold a pencil vertically over the graph, and see if any y's are on the same line.
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