UNIT ONE REVIEW

Scott Simigian

Topics

This will go over all the different topics in unit one of Algebra 2! This will be important for honing in on how to solve or graph these certain topics, as well as connecting them to real life scenarios!


ALGEBRA ONE REVIEW

POINTS TO UNDERSTAND:



Slope intercept- slope intercept is a simple for of an equations. You will always use this format when graphing equations**. Also, it is very useful in solving systems, especially when using substitution to solve the equations. This is because one unit (y) is equal to whatever is on the other side of the equations. Simply plug that into the equations and there you go!


Standard form- this form is most common when you are solving systems of equations. Often all of the systems will be in standard form. When using elimination techniques, make sure to have both systems in standard form to eliminate your was to a solution!


So you make be asking: What even are these two mysterious equation forms?!


Slope intercept- y=m(x)+b

Standard- a(x)+b(y)=c


Let me do some explaining:


Slope intercept is is when you have your 'y' value. The 'm' is simply the mathematical representation for slope. Also, 'b' is a representation for the y-intercept (hence the name slope intercept).


Knowing these equation forms is just the beginning of their potential. These equations forms have everything to do with systems of equations, absolute value, and piecewise functions (slope intercept form)

Check out the video below in order to get a better idea about theses two equation forms!

(Go ahead and learn about point-slope format as well)!


5 Minute Math: Writing a linear equation in 3 forms

DOMAIN AND RANGE

Domain and Range are two fairly simple concepts once you've mastered the fundamentals. For this specific topic, I have created an Educreation to help explain domain and range.
Domain and range have direct relations with piecewise functions and absolute value. Domain is exactly what. Is restricted in piecewise functions which makes the function unique. The range is reliant on the domain in the way that which equations are used for specific x-values


Every single function follows one specific rule. The vertical line test is used to tell whether an equation is a function or not. The vertical line test means, that if a vertical line were to be passed along the entire function, would it touch only one point at a time. If the vertical line touches more than one point at a time then the equation is not a function. In other words, in order for an equation to be a function, 'x' values cannot repeat, but 'y' values can repeat.


Linear functions are your basic equations. These are simple, straight graphs that are constant and continuous, unless restricted of course.


Linear equations are found in all forms of equations (standard form, slope intercept form, and point intercept form)


These are easy to graph

1. Get equations into slope intercept form

2. Plot a point on the y-intercept (remember this is 'b' from the slope intercept form).

3. Graph the rest of the points based on the slope of the line

The only other type of graphs would be a vertical graph. This graph, however, is not even a function (does not pass vertical line test). In that case you would graph the equation. The equation would be 'x=___'



So far, the only piecewise functions have been linear graphs which makes for easy domain and range understanding. Also absolute values are entirely linear graphs, just with restrictions.

Piecewise Functions

Check this slide show from Haiku Deck to see basics of piecewise functions

More information...

Domain and Range:



Whenever you have a restricted domain or range, this will automatically change how you write your domain and range, in either set or interval notation.


Notes- remember the hard and soft brackets from the Educreation. This are very important when putting the domain and range into interval notation.

An easy way to remember when to use hard or soft brackets is whenever the solution INCLUDES the number. Whenever the solution does NOT include the number.


Ex. y<3 (-infinity, 3)

y< or = 3 (-infinity, 3]


Because of this, domain and range are basically made for each other in the fact that piecewise functions are restricted by their domains. This, in turn depicts what the y-value will be.


Absolute Value

Absolute value is the distance from zero, or any specified point.


It can seem a little tricky at first, but when you break it down it is not very hard.


Absolute value is always positive, because distance is always positive. Like the way that 2 is 2 units away from the origin, so the point would be (2,2). Also, since -2 is also 2 units away from the origin, so the point is (2,-2). This is what gives the graph it's 'v' shape.


How to graph:

F(x)= -3|x-2|+5



  1. Start with the origin (0,0)
  2. Anything in the absolute value, with the 'x', do the opposite to the 'x' value. So x-2 means add 2 to the x-value. (2,0)
  3. Anything to the right of the absolute value means do exactly that to the y-value. So add 5 to the y-value. (2,5)
  4. Anything to the left of the absolute value is the slope. So 3 is the slope.
  5. If the slope is negative the 'v' shape of the graph will go down. If the slope is positive, the 'v' shape will face up.


How to solve:

12= 3|x-2|-6





  1. Isolate the absolute value. So add 6 to eliminate the -6. Then divide by 3 to eliminate the 3. If the absolute value is equal to a negative number, then it is no solution. But sense this case has a positive number equal to the absolute value, then continue to solve
  2. Now create 2 equations. Create one the exact same as the equation with the isolated absolute value. Make the other equation contain the absolute value, but flip the other side to its opposite. If the equation is greater than or less than, make sure to flip the symbol when you make the second equation.
  3. Solve the equations as normal and get two answers. Plug the answers in to see if they work.


An easy way to think of absolute value is to think of it as two linear equations that are restricted at their intersection. That is what they are, basically a piecewise function that is equal on each side.




Tiger Woods

Tiger Woods, a famous golf professional, is highly decorated and has won many tournaments. In his better days, (not recently), he finished a championship win with a signature fist pump. Over the years he has made this celebration iconic and wow, it also looks like something we know from math!
The fist pump equation is..... F(x)= |x-2|-2!!!!


Systems of Equations

Systems of equations are two or more equations including at least two variables. These equations are typically in standard form, but also can be useful when in slope-intercept form.




Video Explanations

Here is a video showing how to solve systems of equations based on graphing, putting the systems into tables, eliminating, and substitution


How to Solve Systems of Equations
And one more video showing different uses for systems in the real world and how to set up such situations


Below that are some helpful practice problems!

Special Applications of Systems of Equations
Systems are always in either standard or slope intercept form so those will directly tie into each other. For the most part they are linear equations so it is easy to graph and solve. In case there is an absolute value or a quadratic in a system, you can substitute those for an individual variable, then after you solve for the variable, can make the conversion from the variable back to the absolute value or quadratic. This will make those equations much easier to solve by putting them into a linear format.


THE END