Something I found interesting this week is that equations can be represented in so many different ways. For example you have ordered pairs, table of values, arrow diagram, mapping notation, function notation and much more. To show what some of those listed would look like, we can use 3x+4 as an example.
Ordered pairs: (1, 7) (2, 10) (3, 13) etc.
The x in the example equation represents the input value, the first number in each ordered pair (which can also be any number you choose). What you get after solving the of the equation (in this case multiplying the input by 3 and adding 4) is the y value or output. You can then write the input and output numbers in brackets which give you ordered pairs. These are good if you want to graph the numbers on a cartesian plane.
Table of values:
X | Y
The table of values has the same numbers as the ordered pairs, only instead of matching the corresponding x and y values together in pairs, you would write all the x values together and all the y values together. This is good if you want to find the rule (3x+4) and if you want to find x intercepts (the x value when y is equal to zero) and y intercepts (the y value when x is equal to zero)
Function notation: f(x)=3x+4
For function notation, you can take the general rule (3x+4) and state the name (f) and input value (x) at the beginning. To be more specific with numbers, you can write: f(1)=7 where 1 is the input and 7 is the output. Function notation is good when you want to show that the relation is a function (only one input value per output).
There are many more ways of representing equations, each meaning the same thing but being written differently. They are critical to solving different types of problems and in different situations.
This week we learned about function notation. It is a specific way of writing functions, which is a special type of relation where every x value only has one y value.
Written out, function notation looks something like this:
In the equation F is the name of the function and x is the input value. You could input a number into the equation and you would get a specific answer after you solve it.
To write a function from a table of values where, for example the x values are 1, 2, 3 and the y values are 5, 7, 9 you would follow these steps.
First, find out the rule. To do that, select a pair of x and y values to work with (let’s say 1 and 5). Then take the difference between the each y value (for example in the equation the y values increase by 2) and multiply that by the x value you chose (2×1). Then take the new number and see what you would do to it to get the original y value chosen (2+?=5). After that, write it out as an equation: (y=2x+3)
Now that you have the rule, you can write it in function notation. Write the name of the function, (usually f) then write x in parentheses to represent an input value. Finally write out the rule (2x+3). The final form should look like this: f(x)=2x+3.
This week we reviewed how to graph inequalities on a number line. For example, if you had the equation x > 2 this is how you’d display it properly.
1. Take the number(s) in the equation and find them on the number line. In this case the number is 2.
2. Look at the symbol(s). In the equation, it states that x is larger than 2, using the > sign.
3. Draw either a filled-in circle or a hollow circle on the number. The filled-in circle means that x is greater than or equal to/ less than or equal to the number. The hollow circle means that x is either greater or less than the number. In the example equation, x is greater than 2 so you would put a hollow circle around the 2.
4. Draw an arrow stating if x is greater than or less than the number. In the equation we know x is greater than 2 so you would need to draw an arrow pointing to the right of, and along all the numbers greater than 2.
The end result should look like the picture above.
This week I learned the difference between discreet data and continuous data. Discreet data usually applies to certain numbers of objects or things that are being measured. They are usually things you can count and can not be split into pieces. Some examples of discreet data would be the number of cars in a race, number of marbles in a jar or number of people at a concert. It would not make sense to have half a person, so you can tell it is an example of discreet data. When graphing discreet data using a cartesian graph you would not connect the dots to show that the information is discreet.
Continuous data however usually applies to things that can be measured. They can be measured at different intervals and in different ways. Some examples of continuous data would be the amount of time it takes to complete a task, the distance an airplane travels or the height of a building. It is possible to have one and a half meters, which is how you can tell this is an example of continuous data. When graphing continuous data you would connect the dots to show that the information is continuous.
This week we learned how to use the box method to factor trinomials that didn’t have a leading coefficient of 1 (ugly trinomials). For me, the box method is nice because it doesn’t require me to do as much thinking in my head as the inspection method.
As an example, if you have the expression these are the steps to factor it.
1. Draw a box and split it into quarters so that there are four spaces inside.
2. Write the first term () in the top left-hand corner and write the last term (10) in the bottom right-hand corner.
3. Multiply the two numbers without the variable (30) and find the factors of that number that add up to the middle term (11).
4. Look which numbers already inside the box have corresponding factors or things in common and place them in the two leftover spots. After, add in the variable to the numbers.
5. Find out which numbers multiply to get the numbers in the box and put them on the outside spaces accordingly. It works like a multiplication table.
6. Once you’re done, write the answer in parentheses (3x+5)(x+2)
This week we learned how to factor polynomials. This means that you would take an answer to a FOIL question and find out what the question was. For example, if you take you would first write down all the factors of 49. After that, find the pair of factors that adds up to 14 (7×7). Finally you would write the answer as two binomials (x+7)(x+7).
If you have an expression with two terms () that means that if it were factored, the expression would be conjugates. For example, (x+5)(x-5) are conjugated because the two constants have opposite signs (+ and -). This also means that if you multiply out the expression the +5 and -5 would cancel out and become deep pairs.
This week I learned how to solve problems using double distribution. If you have a question, for example: (2x-3)(2x-3) instead of multiplying the numbers inside the brackets seperately and then multiplying the two numbers left over, you would need to multiply each of the numbers in the first brackets with the ones in the second. You would also have to multiply it in this order: First numbers, outside numbers, middle numbers then the last numbers. The steps should look like this:
- First numbers: Multiply 2x and 2x
- Outside numbers: Multiply 2x and -3
- Middle numbers: Multiply -3 and 2x
- Last numbers: Multiply -3 and -3
When you put it together, it should be -6x-6x+9
Then you can simplify it to -12x+9