Lesson 1-1 Linear Relations and Things related to linear functions Secondary Math III Lesson 1-1 Linear Relations and Things related to linear functions

Vocabulary Relation: A “mapping” or pairing of input values to output values. Anything that can be written as a set of ordered pairs. Function: A relation where each input has exactly one output.

Group the following words into two groups Independent variable Input x-value Range Dependent Variable Domain y-value x value input domain Independent variable y value output range dependent variable

Name 5 ways to show a relation between input and output values. Ordered Pairs: (2, 4), (3, 2), (-4, 3), etc. x 2 3 -4 y 4 Data table: Graph: Equation: y = 2x + 1 Mapping input output 2 3 -4 2 3 4

Identify the Domain Your turn: 2. (2, 4), (3, 5), (-4, 2) 3. 4. 5. x 6 9 -2 y 4 7 3 3. 2 3 -4 -5 4. 5. input output -3 1 3 2 3 -4 -5 2 3 4

Parent function: The most basic function in a family of functions. For lines: y = x is the “parent function”

Which data set is linear Which data set is linear? (For a data set to be linear, the change in the x values and y values are constant) C A B x f(x) 1 2 1.4 3 1.7 4 2.0 5 2.2 6 2.4 7 2.6 8 2.8 9 x f(x) -4 -7 -3 -5 -2 -1 1 3 2 5 7 4 9 x g(x) -4 32 -3 18 -2 8 -1 2 1 3 4

What is the equation that fits the linear data? x f(x) -4 -7 -3 -5 -2 -1 1 3 2 5 7 4 9 What is this number? The value of ‘y’ when x = 0.

Your turn: What is the equation that fits the linear data? x f(x) -4 -7 -3 -5 -2 -1 1 3 2 5 7 4 9 What is this number? Slope:

Your turn: What is the equation that fits the data? x f(x) -4 -9 -2 -6 -3 2 4 3 6 8 9 10 12 15

Discrete Discrete Continuous A continuous relation is a smooth curve or line, a discrete relation has distinct points. Determine if the following relations are continuous or discrete. 3 3 x f(x) -2 -3 -1 1 3 1 1 1 1 Discrete Discrete Continuous

What is the domain and range of each? 3 3 x f(x) -2 -3 -1 1 3 1 1 1 1 Continuous Discrete D = {-2 ≤ x ≤ 1} D = {x = -2, -1, 0, 1} R = {-3, -1, 1, 3} R = {-3 ≤ x ≤ 3}

How do you graph? Check on the graphing calculator Graph y = (2/3)x – 1 and x – 5y = 25

Write an equation in slope-intercept form for the following lines. 2x – 3y = 12 Runs through (-1, 4), slope = 2 Runs through the points (3, -2) and (-1, 4)

What is a sequence? Sequence: an ordered progression of numbers (a list) 5, 10, 15, 20, 25 Finite sequence 2, 4, 8, 16, 32, …,2ⁿ,… Infinite sequence “32,…, 2ⁿ” means the numbers of the pattern continue between the two given terms. “2ⁿ,…” means the numbers of the pattern continue forever. “2ⁿ,…, 264” means the numbers of the pattern continue until the last term in the sequence (264)

How is a sequence a relation between input and output? -9, -6, -3, 0, 3, 6, 9, 12, 15 What are the input values?

The range of a sequence is the actual sequence of numbers. The domain of a sequence is almost always the “natural numbers” k 1 -9 2 -6 3 -3 4 5 6 7 9 8 12 15 The input value is the position in the sequence (1st, 2nd, 3rd, etc.) The range of a sequence is the actual sequence of numbers. The output values are the individual numbers of the sequence.

Represents the sequence named “a” which has subscript “k” to identify the “kth” term of the sequence. Defines a “rule” so that you can find the “kth” term = We call this method of defining the sequence “set-builder” notation. Spoken: “the sequence ‘a’ is defined as 2k +3 with ‘k’ taking on the values 1, 2, 3, and so forth”

Let’s make a table of values that contains the input/output pairs for the first 6 numbers of sequence ‘a’. = k 1 2 3 4 5 6

This is how we write a linear equation that contains the ordered pairs in the table below. = Slope= k 1 2 3 4 5 6

Your turn: Write a linear equation that contains the ordered pairs in the table below. = x 1 2 3 4 5 6

Arithmetic Sequence: a sequence where there is a “constant difference” between each of the adjacent terms. = k 1 2 3 4 5 6 Arithmetic Sequence: each pair of adjacent terms has the same “common difference” Note: the “common difference” becomes the slope of the linear equation passing through each ordered pair.

= How is a sequence similar to a linear function? k 1 2 3 4 5 6 = 5 11 7 13 9 15 5 11 7 13 9 15 How is a sequence similar to a linear function? How is a sequence different from a linear function?

Your turn: write the equation that contains the x-y pairs in the table. f(x) 1 -9 2 -6 3 -3 4 5 6 7 9 8 12 15 Your turn: Define the following sequence of numbers using “set builder” notation. -9, -6, -3, 0, 3, 6, 9, 12, 15

Two ways to write a formula to define the numbers in a sequence. Explicitly Defined Sequence Recursively Defined Sequence These two methods can be used for all sequences, not just arithmetic sequences.

Explicitly Defined Arithmetic Sequence Where (and k = 1,2,3,…) In explicitly defined sequences, you can calculate the “kth” term directly  you don’t have to find any term before it in order to find the “kth” term. Your Turn: What is the 3rd term of this sequence?

Examples Sequences Where (and m = 1,2,3,4) Running total: if your car payment is $200/month, a running total would look like: Month 1 2 3 4 Total ($) 200 400 600 800 Your Turn: 1. Is this an arithmetic sequence? 2. Define this sequence explicitly and name it “C”. Where (and m = 1,2,3,4)

Recursively Defined Sequences have three parts. For all n > 1 Specifies the initial element of the sequence. Gives a way to show the relation between adjacent elements of the sequence. ‘n’ acts as a counter for the sequence

Recursively Defined Sequences For all n > 1 Your turn: Find the 3rd term of the sequence. In order to find the 3rd term, we need to find out what the 2nd term is: 4, 6, 8, 10, 12,… Your turn: Is this an arithmetic sequence? Your turn: What is the common difference? Your turn: What is explicit formula for the sequence?

Your Turn: Find the 4 term of the following recursively defined sequence. For: n ≥ 2 Is the sequence arithmetic? How can you tell from the sequence definition above?

Your turn: Define the following sequence of numbers recursively. -9, -6, -3, 0, 3, 6, 9, 12, 15 For 1 < n ≤ 9 Explicitly defined:

“Generalized” formulas for Arithmetic Sequences -9, -6, -3, 0, 3, 6, 9, 12, 15 Recursively Defined: For all n > 1 d = common difference Explicitly Defined: For all n ≥ 1