In Math 10 this week we stated a new unit called Relation and Functions. Relations are when the the independent variable, x, has multiple connections to dependent variables, y. Functions are when each independent variable has a special connection to one dependent variable.

Often written as f(x), if f(x) = 3x + 5

f(9) = 3(9) + 5

f(9) = 32, where the x variable, 9, has the pair of 32 as the y variable                                      on a graph. This would be a point of the graph. Another way of putting                                      this in words is that 9 is the domain and 32 is the range.

Domain is the set of all the numbers for the independent variable in the relation or function.    Range is the set of all numbers for the dependent variable in the relation or function.            So if we were given the domain { -2, -1, 0, 1, 2 } and the function f(x) = 3x^2 -4 then we can find the range { 8, -4, -1} because all we have to do is replace x in the expression with each number of the domain individually and solve.

These have a connection to the last unit we learned with expressions because often there are inputs that are integers and the output are expressions we were introduced to before.

This week in Math 10 we learned how to factor ugly polynomials (polynomials that aren’t able to be factored with other patterns). There is a simple process to follow and finding the factors of ugly polynomials.

Expression: 12x^2 + 17x + 6

Step 1- Go through the list of patterns and choose which one relates to your expression

Common – any common factors within the expression

Difference of Squares – a squared number subtracted from another squared number

Pattern – does it follow the pattern of polynomial that can be factored (x^2 + x + #)

Easy- no leading coefficients

Ugly- don’t follow any of the patterns

Step 2- Take the leading coefficient and the constant of the expression, multiply them together, and find all the possible ways it can be the product of different numbers.

Step 3- Make a box with 4 squares and put the first term in the top left square and the constant in the bottom right square. 

Step 4- Go through the list of the numbers you created and find the one that equals the middle term. If the sign before the constant is a negative then the pairs are different signs. If the the sign before the middle term is a negative, the bigger number of the pair is a negative and if it is a positive then the bigger number is positive. If the sign before the constant is a positive that indicates that both numbers are the same sign.

Step 5- Put the bigger number and its sign in the top right square and the smaller number in the bottom left. With the box completed, go around the box and take out the Greatest Common Factor (The highest number that divides exactly into two or more numbers) with each row and column. The length and width of the box is the factored form of that expression. 

This week in Math 10 we started to learn how to factor polynomials which means to break down polynomials into simpler terms so that when they are multiplied together, they equal the original polynomial.

Step 1: In this example that I have created I started off by finding the greatest common factor (the biggest number that divides into 

the given numbers) of 14, 21, and 35. I did this buy finding the prime factorization of these three numbers and found the common number which is 7.

Step 2: Then I looked for the greatest common factor of the variables and their exponents by looking at which variables I had which were a and b. Then I looked at the smallest exponent of these variables which were 2 for both a and b

GMF of this equation is 7a^2b^2

Step 3: Then I wanted to be able to use this gmf to equal the original polynomial so I had to find out how many times to multiply this gmf to equal all the terms

I ended up with 7a^2b^2(2+3a-5b) 

This week in Math 10 we were introduced to polynomials, but only reviewed what we were taught in grade 9. Some things I remember are models, degrees, types, and distributive property,

Example: 5(8x – 3y) + 2(4y + x)

Step 1: Multiply 5 by 8x and then again with -3y

Ex. 40x – 15y + 2(4y + x)

Step 2: Mutiply 2 by 4y and then again with x

Ex. 40x – 15y + 8y + 2x

Step 3: Collect like terms. Add 2x to 40x and add 8y to -15y

Ex. 42x – 7y

FInal Answer- 42x – 7y

This week in Math 10 we learned trigonometry and other aspects such as finding missing side lengths using trig.

Step 1: first step to any trig equation is to label all the sides and make sure theres a number on the inside and outside unless it wouldn’t work to use trig.

Step 2: Find out which variables you are given and figure out which side they belong to (ex. hypotenuse, opposite, or adjacent)

Step 3: Use the equation ratio= side 1 divided by side 2

Step 4: If the variable is on the bottom, reciprocate both sides of the equation

Step 5: Since we want to isolate x, multiply the bottom of the fraction with x and multiply it to both sides.

Step 6: Solve

FInal Answer: x = 2.6

This week I learned about conversion between the SI system and conversion between the Imperial System.

First step to convert between the imperial system is take 75 inches and put it over one.

Step 2: Take a conversion of inches to feet which would be either 12inches over 1foot or 1foot over 12 inches. I chose 1foot over 12 inches because the inches were diagonal from each other

Step 3: Cross out the inches because the two cancel each other out being diagonal from each other.

Step 4: Finish the equation which left me with 75 over 12 and divide 75 with 12 and the answer is 6.25.

6 will be the number of feet.

Step 5: Take .25ft and put that over one and this time multiply it by 12 inches over 1 foot because the feet have to be diagonal from each other.

Step 6: Cross out the ft because they cancel each other out

Step 7: Finish the equation which left me with 3 inches

Final Answer: 6feet 3inches