Week 16 Math 10

This week in math 10 I learned about the different sequences and how they are classified. The three main types of sequences are arithmetic sequences, which are the ones we are focusing on most in this unit, then the geometric sequences, and then any other sequences that do not fall into either of those categories. Within each of these sequences they can be either finite which is when the sequence has an end or will eventually stop or they can be infinite, when they do not have a stop and that is represented by … at the end of a sequence.

Arithmetic Sequences 

Arithmetic sequences are when the pattern or common difference of the sequence is the same number adding or subtracting to each term.

Example:

 

 

 

 

Geometric Sequences

Geometric sequences are when the sequence is multiplied by the same number to each term.

Example:

 

 

 

 

 

Other Sequences

The other sequences is any thing that is not a geometric or arithmetic sequences.

Example:

 

 

 

 

 

 

The pattern in the example is the two numbers in front are added to get the next number.

You can also take these sequences and put them into a chart by putting the sequence into the y and the number of each term in the x column. If the sequence is arithmetic the common difference of the sequence (the number that is added or subtracted from each term) when you graph it you will notice that it is the same as the slope of the line.

If the table is

X      Y

1        2

2       4

3       6

4       8

5      10

The common difference between the y values is 2 on the table and if you look at the graph below you can see that the slope of the line is 2.

Week 15 Math 10

This week in math I learned another way to find the point where two lines cross, which is called elimination. We also learned how find two equations when given a word problem and then with the equations solve the system to get the answer to the problem.

Elimination 

For you to be able to to find the point you need to make sure that the equations line up, so it they are not then rearrange them and it is important to remember that when a term goes to the other side of the equals sign it changes to the opposite sign that it had.

 

 

 

 

Once the equations are lined up you need to make sure that either the x’s or y’s cancelled each other out or create a zero pair (the variables have the same coefficient except one is positive and the other is negative so they cancel out each other), if there is not one already you need to multiply one or both equations by a number that will make there be a zero pair.

 

 

 

Then add the like terms of both equations and then you need to isolated the variable.

 

 

 

 

 

 

 

 

Now with the variable that you have you plug that into either of the original equations to find the variable that you do not have.

 

 

 

 

 

 

 

 

Then you have your answer and you can verify it to make sure that you are right.

 

 

 

 

 

 

 

 

Word problems 

Example: The difference of two numbers is 3. Twice the larger number is 28 more than the smaller number. Find the numbers.

First it is important to read the word problem over slowly, then again but this time highlight the important words.

The words that are bold, are the ones that will help to make the equations, each sentence is one of the equations. Difference is subtracting two numbers. Is represents the equals sign. More is adding a number. Than means to switch what is in front of it and what is after it.

So, when finding these equations it is important to determine which missing number is x and which missing number is y. Then using the hints from the words create your equations.

 

 

 

Once you have the equations you can now solve using substitution or elimination.

 

 

 

 

 

 

 

 

Lastly verify your answer.

 

 

 

 

 

 

 

Week 14 – Math 10

This week in math 10 I learned what systems of lines are and how to find the point at were they cross by using graphing and substitution. Systems of lines are when given 2 equations and you have to find were they cross.

Using Graphing

To find were they cross you need to rearrange the equation into slope y intercept form (y=mx+b). When doing this you need to make sure that the y is positive and by itself with no coefficient and made sure when putting something on the other side of the equals sign.

 

 

 

 

 

 

 

From looking at just the rearranged equations you can tell how many solutions (how many points the two lines will cross). If the equations have same slope but a different y intercept, they are parallel so they will never touch, which means there is no solutions. If the slope is different the lines will cross and there will be one solution. if both the slope and y intercept are the same then there is a lot of solutions because they are on top of each other.

 

 

 

 

 


Using Substitution

When given two line equations substitution is another way to find a solution. When looking at the equations you need to choose one of the variables and isolate it.

 

 

 

 

 

 

Then put the new rearranged equation into the other one that you did not rearrange. Next you next to solve for the single variable. Then depending on which you isolated first you now either have the x or y value.

 

 

 

 

 

 

 

 

With the variable you have plug that into the rearranged equation to find the other missing value.

 

 

 

 

 

 

 

 

Verifying Solution

Once you have the two coordinates of where the lines cross there is a way to check your answer by plugging in the x and y values into the equations if the numbers on both sides of the equals sign are equal for both equations then you know you have the right answer.

 

 

 

 

 

 

Week 13 – Math 10

This week in math 10 I learned the three different ways of writing equations for straight lines, how to find the slope, y-intercepts and how to move and rearrange terms to change one equation into another. The three equations are slope y-intercept form, point slope form, and general or standard form.

The first equation is slope y-intercept form which is written as y=mx+b. This equation makes it very easy to find the slope and y-intercept. The y is the output, the x is the input, in front of the x is the slope represented by m, and b represents the y-intercept.

An example is trying to find an the slope y-intercept equation when given another equation that has parallel or perpendicular slope and given the y-intercept.

 

 

 

 

The next equation is point slope form which is very useful when given a point on the graph and the slope. The equation is written as m(x-x)=y-y.

An example of getting a point (-2, 5) an a slope of 3, then putting that into point slope form. To put this information into the point slope form equation, the slope goes in place of the m in front of the x. the -2 is the x spot so you put that into the second x and you add it because two negatives equal a positive. The 5 is in the y spot so you put it in place of the second y and that is your point slope form equation.

 

 

 

The last equation general form which is written as Ax+By+C=0. Only integers can be used in this equation no fractions the leading coefficient (value in front of the x) has to always be positive. This equation is  good for finding both the x and y intercepts.

Finding the slope, x, and y intercepts when given an equation in general form.

To find the slope you need to rearrange so that you have it in slope y-intercept form making it easy to identify the slope. In order to do this you need to move the y or everything else to the other side of the equals sign so that the y stays positive.

 

 

 

 

 

 

 

 

To find the x and y intercepts depending on which one you are trying to find you put zero in place of the x or y value. Then isolate the left over variable.

 

 

 

 

 

 

 

 

When trying to convert from one form to another it is important to know which ones are possible to rearrange into each other and which ones are not.

 

Wonky Initials

 

 

 

 

 

 

 

 

The red lines are the lines that I also used the slope y-intercept form equation to represent the line as well as the point slope form and the black lines are the lines that I also used the general form to represent the line equation as well as point slope form.

S

Table

X   Y

-5   13

-8    6

-9    -1

-10   17

-14   5

-15   10

-16   3

-17   15

Equations

 

 

 

 

 

 

 

 

 

 

 

J

Table

X  Y

5   3

1    6

1   -2

-3  -8

-5   9

-6  -3

-7  -6

Equations

 

 

 

 

 

 

 

 

K

Table

X   Y

16   -4

13  -11

10  -15

9      1

7    -4

5   -13

 

Equations

 

 

 

 

 

Week 12 – Math 10

This week in math 10 I learned how to find the distance and mid point between two boundary points of a line segment. First it is important to know the difference between a line which is straight and on going and a line segment which is a straight line with two boundary points.

To find the distance between two boundary points of a horizontal line segment, it is important to know that horizontal line segments will always have the same y value and that distance it subtracting two numbers if both are positive or negative, if on is negative and the other is positive then add the numbers. Then you need to know what numbers to subtract which are the x values or the values that are different.

 

 

 

 

 

 

 

To find the mid point of a horizontal line segment, you need to add the x values then divide them by two to get the x value of the middle point and the y value is the same as the two boundary points.

 

 

 

 

 

 

 

To find the distance between two boundary points in a vertical line segment it is the same as horizontal line segments except the x values are the same and the y values are different. To find the distance you are subtracting two numbers if both are positive or negative, if on is negative and the other is positive then add the numbers. Then subtract the y values to get the distance.

 

 

 

 

 

 

 

To find the mid point for a vertical line segment it is the same as a horizontal line segment, add the two y values then divide them by two and the x coordinate is the same as the boundary point x values.

 

 

 

 

 

If the line segment is oblique which is when it is diagonal and both the x and y values are different you need to find the horizontal length and the vertical length. To find the horizontal length you need to subtract the y values, then to find the vertical length subtract the x values.

 

 

 

 

 

 

 

Once you know how the lengths if you look on a graph it forms a right triangle and your only missing side is the hypotenuse so you need to use the Pythagoras theorem (a^2 + b^2 = c^2) using the horizontal and vertical lengths. The answer is your Distance.

 

 

 

 

 

 

 

To find the mid point of an oblique line segment you take both the x values from the boundary points and add them then divide them by two for the x coordinate of the midpoint. To find the y coordinate for the mid point you need to do the same thing as the x values, by adding the y values then dividing them by two. Then you have the coordinates for the midpoint.

 

 

 

 

Week 11 – Math 10

This Week in math 10 I learned how to interpret graphs of functions. When interpreting the graphs it is important to know what the x and y axis are in the situation so that you can have a better under standing of how the line is going to look on the graph. If you are given a statement describing the line on the graph and given options of which line is best, it is important to know that the steepness of the lines are very important, when a line is more vertical the steepness is higher because it is going up faster, so more vertical lines are usually used when someone is running or biking. When the line is horizontal then for example if we are talking about a person and they are not moving they are staying in the same place then the line will be horizontal or if the line is more horizontal then vertical it means that the line is less steep making it go up slower, for example it could represent someone walking.

Example: The line on the graph shows a girl coming home from school. It shows the time that she has traveled a certain distance from school.

Week 10 – Math 10

This week in math 10 I learned how to what was special about a relation to make it a function, how to solve a function equation and what each part in the equation means.

Functions

To separate a function from any other relation, it is important to know that in a function the input number can only have one outcome or one output number, however if two different input numbers have the same out put that is still a function.

 

 

 

 

 

 

When reading a graph to tell if it is a function equation you look to see if there are two or more points directly above each other they are not functions they are relations.

 

 

 

 

 

Function Notation

First it is very important to know what each part of the equation means. The letter in front of the equation is the name of the equation (usually f, g, h, or j), the x in brackets is the input number, and after the equals sign is the function equation.

 

 

 

 

When given the value of x the you put that number in place of all the other x values in the function equation, then solve.

 

 

 

 

 

 

If given the output number then you need to solve for x by isolating the variable. first get ride of the constant by adding or subtracting, then multiply of divide the coefficient, and if the x is squared then you need to square root to get the answer (make sure to realize that there could be more then one answer to the square root).                                                                                                                                     

 

 

 

 

 

 

 

 

If given an equation with both x and y on the same side of the equals sign and a zero is on the other you need to get the y alone on one side so, start with the constant by adding or subtracting it from the zero, then add or subtract the coefficient and variable to get them on the other side as well so that the y is by its self.

 

 

 

Week 9 – Math 10

This week in math 10 I learned what domain and range is and how to find them when given a graph with a line or if there is just dots on the graph. First know what inequalities are and how to write them is very helpful when you then move on to finding domain and range.

Inequalities

If the arrow is pointing to right (toward the positive numbers) you use the greater sign and when arrow is pointing left (toward the negative numbers) you use the less than sign. If the boundary point is opened then it is just either less than or greater than, but if the boundary point is closed then it is less than greater than and or equal to.

 

 

 

 

 

 

 

When there is two boundary points you need to find the two inequalities then combine them by putting the signs the same way and make sure to pay attention if the boundary point is opened or closed.

 

 

 

Domain – Is the set of all the numbers for the independent variable (x-axis) in a relation.

Range – Is the set of all numbers for the dependent variable (y-axis) in a relation.

When finding the domain with a graph with only dots you find what value each of them are on the x-axis then to show that they are a set you have to use these brackets { }, then write the values from least to greatest and if a value shows up more then once, only write it once.

 

 

 

 

 

 

 

 

When finding the range with a graph with only dots, you need to find the values of the dot on the y-axis and then you do the same thing to show it as you did for domain.

 

 

 

 

 

 

 

 

When finding the domain on a graph that has a line you need to find the boundary point that is smaller on the axis, then find the other boundary point and what the value of it is on the x-axis (make sure to pay attention to the boundary points if they are opened or closed).

 

 

 

 

 

 

 

Now that you have your values you need to put the signs in between facing the same way.

 

 

 

 

 

 

The last thing that you need to do is because it is a line it counts all the values in between so you need to write x is an element of the real numbers, to show that it is a line.

 

 

 

 

 

When finding the range on a graph that has a line it is very similar to domain, but you need to find the boundary that is the highest value, so x will be less then.

 

 

 

 

 

 

The last thing that you need to do is because it is a line it counts all the values in between so you need to write x is an element of the real numbers, to show that it is a line.

 

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