UNIT ONE REVIEW
ALGEBRA ONE REVIEW
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
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)
(Go ahead and learn about point-slope format as well)!
DOMAIN AND RANGE
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
Linear Parent Function
Most basic, slope=1, y-intercept=0
Negative Linear Function
Flip the line over the y-axis, slope=-3/4, y-intercept=3
Horizontal Linear Function
Horizontal graph, slope=0, y-intercept=-2
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.
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.
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:
- Start with the origin (0,0)
- 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)
- 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)
- Anything to the left of the absolute value is the slope. So 3 is the slope.
- 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:
- 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
- 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.
- 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.
Systems of Equations
Below that are some helpful practice problems!