Chapter 2 Unit Overview

Chapter 2 Unit Overview

1. Functions

   • A relationship for which each value of the control (independent variable) is paired with only value of the dependent variable
     • One to one function
       • "One x has one y"
     • Many to one function
       • "All x's have one y" or "some y's have multiple x's"
   • Independent Variable: A variable that controls or determines another variable
   • Dependent Variable: A variable that is determined by another variable
   • Domain and Range
     • Domain: All possible values of the independent variable (x)
     • Range: All possible values of the dependent variable (y)
   • Function Notation
     • When a variable is a function of another variable
     • Ex: F(x)= 5
       • "F of x equals 5"
   • Linear Function
     • Notation: f(x)= mx+b
     • Domain and Range
       • Domain: All real numbers
       • Range: All real numbers

2. Piecewise and Absolue Value Functions

   • Piecewise Functions
     • A function defined by two or more equations
   • Absolute Value Functions
     • A function that has a variable within the absolute value symbol
     • Can be written in piecewise format

3. Quadratic Functions

   • Common Equations
     • Vertex:
     • Quadratic Formula:
     • Discriminant:
     • Vertex Form:
     • Set Builder Notation:
       • All y's:
       • All x's:
   • Graph is parabola
     • a>0 opens up
     • a<0 opens down
   • Equations for thrown and dropped objects
     • Thrown:
     • Dropped:
   • Notation:

4. Polynomial Functions

   • Characteristics
     • Coefficients are real numbers
     • Exponents are positive
     • Exponents are whole numbers
     • The degree(highest power) is the value of the highest exponent
     • Cannot have roots, x on the bottom (denominator)
     • X cannot be rooted
   • Direct, Indirect, Direct with square, and Direct with cube
     • Direct:
       • y=kx
     • Indirect:
       • xy=k
     • Direct with square:
     • Direct with cube:
     • can be written as function:
   • Cubics
     • K>0 means left to right, increasing
     • K<0 means left to right, decreasing
     • Single, double, and triple zero
       • Single Zero
         • y = (x+1)(x-1)(x+5)
       • Double Zero
       • Triple Zero
   • Zeros
     • Characteristics
       • A value of x that makes f(x)=0
       • A place where the graph of f(x) crosses the x-axis (x-intercept)
     • A solution to the equation (x=1, x=2, etc
     • Change the function or equation to equal 0 and find the zero(s)

5. Radical Functions and rational expressions

   • Radical Function
     • A function that has a variable under a radical symbol
     • How to solve a radical function
       • 1. Move everything but the root to one side
         • 2. Square both sides
           • 3. Rewrite in quadratic notation
             • 4. Solve using quadratic formula or factor
               • Check for extraneous solutions
     • Properties of radicals
   • Rational Expressions
     • Contains a ratio
     • Quotient of two polynomials
     • Must contain variables
     • When the equation is undefined, the x value that is undefined often creates an asymptote
       • Asymptote: a line that the graph approaches more and more closely
       • To find the asymptote, substitute the denominator into an equation equal to 0. Ex: x-2=0 --> x=2 as the vertical asymptote
       • Horizontal asymptote
         • If the degree in the numerator > the degree in the denominator
           • No horizontal asymptote
         • If the degree in the numerator = the degree in the denominator
           • Place coefficient and variable with exponent in the numerator and divide by the coefficient and variable with exponent in the denominator
         • If the degree in the numerator < the degree in the denominator
           • y=0

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