THE BEST CLASS EVER…ERRR…. PRE-CALCULUS Chapter 2 and 4 Midterm Review

Chapter 4 Previous Lesson Materials

What was Chapter 2 about? Graphing Polynomials  Define and recognize a polynomial function  Explain the relationship between end behaviors of a polynomial and its leading coefficient  Predict the basic shape of a polynomial function  Graph a polynomial function Applications of Polynomials  Create a polynomial function to model a given situation  Calculate Maximum and Minimum Piecewise and Step Functions  Define and distinguish between piecewise and step functions  Evaluate a piecewise function  Evaluate a step function  Graph a piecewise function  Graph a step function  Write the equation of a piecewise function from its graph  Write the equation of a step function from its graph  Find domain and range of a piecewise function  Find domain and range of a step function

End Behavior L.C. > 0L.C. < 0 n odd x  ∞, then f(x)  ∞ x  - ∞, then f(x)  - ∞ x  ∞, then f(x)  - ∞ x  - ∞, then f(x)  ∞ n even Rise towards ∞ together Fall towards -∞ together

A linear function is a function of the form f(x)=mx+b The graph of a linear function is a line with a slope m and y-intercept b. (0,b)

A constant function is a function of the form f(x)=b b x y

Absolute Value Function (1,1) (-1,1)

The square function

Cubic Function (1,1) (-1,-1)

Square Root Function

Cube Root Function

Rational Function (1,1) (-1,-1)

Piecewise Function: a) Find f(-1), f(1), f(3). b) Find the domain. a) f(1) = 3 f(-1) = = 2 f(3) = = 0

GRAPH

What was Chapter 4 about? Reflections - General  Describe a reflection about the x-axis  Describe a reflection about the y-axis  Describe a reflection about the line y = x  Describe a reflection about the origin Symmetry - General  Graph a reflection about the x-axis  Graph a reflection about the y-axis  Graph a reflection about the line y = x  Graph a reflection about the origin Function Operations  Evaluate the sum between two functions  Evaluate the difference between two functions  Evaluate the product between two functions  Evaluate the quotient between two functions  Evaluate the composition between two functions Inverse Functions  Define an inverse relationship  Determine when an inverse is a function  Define a composition of functions and its relationship to inverses  Graphically explain the relationship between a function and its inverse  Determine the inverse equation from a function Radical Functions – General  Identify a rational function  Find asymptotes Rational Functions – Graphing  End behavior  Asymptotes  Discontinuities  Domain and Range

Reflections – The Basics x-axisy-axis

Reflections – The Basics y = x Absolute Value

Symmetry – The Basics Symmetry in the x-axis: Meaning: (x, -y) is on the graph when (x,y) is. Symmetry in the y-axis: Meaning: (-x,y) is on the graph whenever (x,y) is. Symmetry in the line y = x: Meaning: (y,x) is on the graph whenever (x,y) is. Symmetry about the origin: Meaning: (-x, -y) is on the graph whenever (x,y) is

Function Operations – Practice

How to create a Inverse

Graphing Rational Functions – The Basics Appears in the following format: Has 2 asymptotes:  x=h (vertical)  y=k (horizontal) In order to graph:  Draw the lines for the asymptotes.  Select two points on each side of every asymptote, plug into your x/y chart and graph.

Complex Rational Functions Appears in the following format: In order to graph:  Draw the lines for the asymptotes.  Select two points on each side of every asymptote, plug into your x/y chart and graph. Asymptotes/Quirks:  Can have multiple vertical asymptotes.  Can have multiple horizontal asymptotes horizontal asymptotes.  Might have holes.

Graphing Rational Functions – Practice Determine the domain and range of the rational function below:

Complex Rational Functions Appears in the following format: Asymptotes/Quirks:  Can have multiple vertical asymptotes. How to Determine VA’s 1. Factor the numerator and denominator. 2. Determine what values would make the denominator equal to 0.

Complex Rational Functions Appears in the following format: Asymptotes/Quirks:  Can have multiple horizontal asymptotes horizontal asymptotes. How to Determine HA’s 1. Look at the degrees of the numerator and the denominator. 2. Follow and memorize the guide on the next slide.

Graphing Rational Functions – Practice Determine the domain and range of the rational function below:

Horizontal Asymptote Guide If the degree of the numerator < degree of the denominator  Then is the horizontal asymptote. If the degree of the numerator = degree of the denominator  Then is the horizontal asymptote. If the degree of the numerator > degree of the denominator  Then there is no horizontal asymptote.

Complex Rational Functions Appears in the following format: Asymptotes/Quirks:  Might have holes. How to Find Holes 1. Factor the numerator and denominator. 2. If there is a common factor in the numerator and the denominator, set it equal to zero. Solve and the value you find is the x-coordinate of the location your hole occurs at.