Unit 1 Algebraic Functions and Graphs
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1 Algebra 2 Unit 1 Algebraic Functions and Graphs Name:
2 Unit 1 Day 1: Function Notation Today we are: Using Function Notation We are successful when: We can Use function notation to evaluate a function This is important because: Function notation is used throughout Algebra 2 Warm up problems a. Evaluate 34 when x = 3 b. Evaluate when x = 2 A is a relation in which each input is paired with exactly one output. The input is what is called the. The output is called the. We can sometimes write equations in function notation. These equations contain the term f(x), which we read as f of x. The letter in front refers to which function we will be using, and the term inside of the parenthesis is what we will be plugging in for our variable. (f(x) is the same as y) Equation Function y = 5x + 3 f(x) = 5x + 3
3 Example 1: a.) Find 3 b.) Find 1 c.) Find 2 d.) Find 11 Example 2: 35 a. Find 4 1 b. Find 2 1 c. Find 3 d. Find 1 e. Find 2 3 f. Find 4 f (7n9) 3
4 Function Unit Day 2: Relations and Functions NOTES Today we are: identifying functions We are successful when: we can decide if a relation is a function using a table, mapping, or graph This is important because: Algebra 2 is based on all different kinds of functions Getting Started: Identify the quadrant or axis the points lie on. Then, graph the points. A: ( 2,3) B: ( 3,0) C: (4, 6) D: (0,5) E: (4,3) Some important definitions: Relation A set of Domain The set of all the independent variables or (INPUTS) of a relation. Range The set of all the dependent variables or (OUTPUTS) of a relation. Function A relation where every element of the is paired with exactly one element of the Mapping an illustration showing how each element of the domain is paired with another element of the range in a relation Ex. #1 Express the relation {(4,3),( 2, 1),( 3,2),(2, 4),(0, 4)} as a table, a graph, and a mapping. Is the relation shown a function? Yes or No Why?
5 Ex. #2 Determine if the following relations are functions. x y A. B. C Vertical Line Test A method of determining whether a graph is a function or not a function If no vertical line can be drawn that will intersect the graph more than once, then the graph is a function. If the vertical line intersects the graph at least twice, the graph is NOT a function. Why does the vertical line test work? Ex. #3 Determine whether the following graphs are functions. A. B. C. D. E. F.
6 Inverse Relations A relation that occurs when the elements of the domain elements of the range. x f( x) with Ex. #4 See mapping to the right. A. Does this mapping represent a function? Why? B. What is the domain of the relation? C. What is the range of the relation? D. Write the inverse of this relation. Practice Problems: #5 See the relation to the right. {(4,3),( 2, 1),( 3,2),(2, 4),(0, 4)} A. Does this relation represent a function? Why? B. What is the domain of the relation? C. What is the range of the relation? D. Write the inverse of this relation. #6 See the relation to the right. A. Does this relation represent a function? Why? B. What is the domain of the relation? C. What is the range of the relation? D. Write the inverse of this relation.
7 Unit 1 Day 3 Identifying Graphs and Functions NOTES Today we are: Identifying parts of function graphs We are successful when: We can look at a graph of any function and identify domain, range, maximum and minimum, end behavior, and x and y intercepts. This is important because: Understanding the parts of a function graph will allow us to determine how certain functions behave. Getting Started: Is this relation a function? List the domain of the relation: List the range of the relation: Write the inverse of the relation. Some important definitions: End Behavior Examines the trend of a function s y values as the x values approach infinity or negative infinity ( x, f( x)?) Infinity A number greater than any assignable or countable number Interval A set of all real number values or two given values. Maximum value The greatest of the values in a function Minimum value The of the output values in a function X intercept The x coordinate of a point where the graph crosses the ; always has an y value of. Where f( x) 0. Y intercept The y coordinate of a point where the graph crosses the ; always has an x value of. Also, denoted as f (0) We found out earlier how to find the domain and range of a relation when they are given to us in various forms, including tables, mappings, and graphs. What happens though when the function you are given is in graph form and it is more than just a couple points? We have to speak in terms of intervals. Ex. #1: Here is a graph of some function y f( x). What is the domain? What is the range? End behavior: As x, f( x) As x, f ( x) f (2) f (0)
8 Ex #2. Here is a graph of some function y f( x). What is the domain? What is the range? End behavior: As x, f( x) As x, f( x) f (2) f (0) Ex #3. Here is a graph of some relation. What is the domain? What is the range? Identify any x intercepts: Where does the minimum value occur? What is the minimum value? Ex. #4. Here is a graph of some function y f( x) What is the domain? What is the range? End behavior: As x, f( x) f (4) f (0) What is the minimum value of y f( x)? Where does the minimum value occur?
9 Ex. #5. Here is a graph of some function y f( x) What is the domain? What is the range? As x, f( x) As x, f( x) f (2) f (0) f ( 3) f ( 1) Identify the x intercept: Identify the y intercept: Ex. #6. Here is a graph of some relation y f( x) What is the domain? What is the range? As x, f( x) As x, f( x) f (2) f (0) f ( 2) f (3) Identify the x intercept: Identify the y intercept:
10 Unit 1 Day 4 Function Intervals NOTES Today we are: Identifying when functions increase or decrease We are successful when: we can look at a graph and correctly describe the intervals when a function is increasing or decreasing This is important because: In Algebra 2, we need to be able to describe how functions behave using mathematical notation Getting Started: Is this a function? Why or why not? What is the domain? What is the range? End behavior:,, 0 2 More definitions A function is said to be over an interval if the graph is going and to the. A function is said to be over an interval if the graph is going and to the. Over what interval(s) is the graph increasing? Over what intervals is the graph decreasing? What is the minimum value? Where does it occur? What is the maximum value?
11 Is this a function? Why? Domain: Range: End Behavior,, Increasing: Decreasing: Is this a function? Why? Domain: Range: End Behavior,, Increasing: Decreasing:
12 Unit 1 Day 5 Graphing Functions Today We are: Graphing functions We are successful when: We can choose points that will help us draw the shape of any graph This is important because: Each unit in Algebra 2 focuses on a different kind of function and to be successful we want to be able to evaluate any function. Looking Back: Is this a function? Why? Domain: Range: End Behavior,, Increasing: Decreasing: The graph of a function is the collection of all input and output values for that function placed on a coordinate plane. While it would be impossible for us to show every possible input and output, we can find a few coordinates and predict the shape of the graph. Domain: Range: Increasing: Decreasing: Y intercept: Minimum:, X Intercept: Maximum:,
13 Domain: Range: Increasing: Decreasing: Y intercept: Minimum:, X Intercept: Maximum:, 1 Domain: Range: Increasing: Decreasing: Y intercept: Minimum:, X Intercept: Maximum:,
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