Parallel Lines are two lines with the same distance apart their entire length, and never touch.
Intersecting Lines are two lines which cross at a specific point, creating an X.
Perpendicular Lines are two lines whose cross creates a right angle.
Basic Functions & Their Graphs:
A Function has exactly one value as the domain, or x, and exactly only one value as the range, or y. A Function is only a function if it passes the vertical line test. The vertical line test is drawing a vertical line straight down the graph of the function in question. If the line meets with the function in question only once, it is definitely a function.
Slope & Point-Slope
To find the slope of a line, or the amount of a non vertical change, use the formula:
y= mx+b
The formula for the point-slope form of an equation, or to find the slope of an equation by using two given points, use the formula:
y – y1 = m(x – x1)
Transformations, Shifts, Shrinks, Stretches, and Reflections
A transformation is a change in the graph of a function, transformations can be either horizontal or vertical
There are 3 types of transformations: translations (shifts), shrinks & stretches, and reflections
1) Translations (shifts)
A shift is when a graphed function moves from one place in the graph to another (either horizontally or vertically) without changing the size or shape of the graphed function
A translation (or shift) is usually the cause of adding or subtracting a number to the function
For example: y= x^2 shifted is y= x^2 +3, or y= x^2 -3
2) Shrinks or Stretches:
Shrinks and Stretches are when the graphed function either becomes smaller (shrinks) or widens along the graph (stretches)
This is usually the effect of multiplying x times a number, or placing a number before it
For example: y= x^2 shrunk is y= -3(x^2). For a shrink, the number infront of x is always negative y= x^2 stretched is y= 2(x^2)
3) Reflections
A Reflection is when a graphed function is completely changed to it's opposite version
The cause of a reflection is multiplying the function by a negative
For example: y= x^2 reflected is y= -(x^2)
Combination of Functions:
Combination of Functions is when one function is included into another one
Usually shown as f(g) or function of g
To solve, substitute g by the function given
For example: let f(x)= 2^x and g(x)= √ x+1 f(g(x))= f(x)= 2^x f(g(x))= 2^g(x) f(√ x+1)= 2^(√ x+1) Use: f(g(x))= (f times g)(x)