Unit 7 - Functions Learning Target : I can determine if a relation is a function. A "relation" is just a relationship between sets of information. A function is a relationship that assigns exactly one output value to one input value.

Unit 7 - Functions Now if we change the number line to people on the x-axis it becomes a function. Everyone has only one birthday. Everyone place your birthday on a number line that lists the days of the year as the x value This is not a function since many days have multiple birthdays

Functions can also be shown as graphs The vertical line test can be used to show if a relation is a function

Vocabulary you should know Function - This is like an equation, one output per one input Domain – Input Value, or x-values Range - Output Value, or y-values Function Table - Organizes the information

Determine if ordered pairs are a function (1, 1) (2, 2) (3, 3) Function (1, 1) (2, 1) (3, 3) (1, 1) (1, 2) (3, 3) Not a Function

Determine if a mapping of points are a function. 1 5 7 3 2 4

Determine if a mapping of points are a function. 1 5 7 3 2 4

Determine if a graph is a function

Determine if a graph is a function

LT: I can represent linear functions Define Linear A line contains at least two points Define Linear Function Line that is a function. There is one line that is not a function, can you name it? Vertical Line

Day 2. Read Page 469 -470. Can you complete the following function table? Given the function f(x) = x + 5 Domain { -2, -1, 0, 1, 2} Range { 3, 4, 5, 6, 7}

Do You know where the domain, function, and range go in the function table? Domain or x Function f(x) = x + 5 Range or f(x) -2 -2 + 5 3 Domain or x Function f(x) = x + 5 Range or f(x) -2 -2 + 5 3 -1 -1 + 5 4 0 + 5 5 1 1 + 5 6 2 2+ 5 7

Can you complete a function table given the function and domain? Given the function f(x) = 2x + 3 Domain { -2, -1, 0 1, 2} This time you must find the range How much do you know? Domain or x Function f(x) = 2x + 3 Range or f(x) -2 -1 1 2

Table filled in -2 2(-2) + 3 -1 2(-1) + 3 1 2(0) + 3 3 2(1) + 3 5 2 Domain or x Function f(x) = 2x + 3 Range or f(x) -2 2(-2) + 3 -1 2(-1) + 3 1 2(0) + 3 3 2(1) + 3 5 2 2(2) + 3 7

Unit 7.2 Graphing Functions Recall how to graph an ordered pair (x, y) The x value is left or right, y up or down

Write down the following Mr. Scott’s Graphing helpful hints 1) f(x) and y are the same values on a graph If a domain is not given use { -2, -1, 0, 1, 2} If there is a fraction with the x, use multiplies of the denominator for your domain. You may have to change the y- axis value so you can put a few ordered pairs on your graph

Example 1 Graph y = 3x + 5 -2 3(-2) + 5 -1 3(-1) + 5 2 3(0) + 5 5 1 Make a table Completed Table x y = 3x + 5 y -2 3(-2) + 5 -1 3(-1) + 5 2 3(0) + 5 5 1 3(1) + 5 8 3(2) + 5 11 x y = 3x + 5 y -2 -1 1 2

Write as ordered pairs and graph (-2, -1) (-1, 2) (0, 5) (1, 8) (2, 11)

Example 2 y = 2 3 x + 1 Make a table with multiplies of 3 and 0 x y = 𝟐 𝟑 x + 1 y -6 -3 3 6

Example 2 y = 2 3 x + 1 Completed Table Graph the ordered pairs -6 𝟐 𝟑 (−6) + 1 -3 𝟐 𝟑 (−3) + 1 -1 𝟐 𝟑 (0) + 1 1 3 𝟐 𝟑 (3) + 1 6 𝟐 𝟑 (6) + 1 5 Completed Table Graph the ordered pairs

Read Page 198 - 199 3 things to consider With your group Read and Answer the Check Your Progress Questions and Check them with me. There is an a, b, and c. You can use a calculator. Change in value = present value – previous value ( moving backwards) Rate of Change = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑎𝑙𝑢𝑒𝑠 𝑡𝑖𝑚𝑒 (𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒) 3 things to consider Unit Rate should have a denominator of 1 Positive or Negative Label both

Unit 7 LT 3: Find the rate of change given a graph or table Rate of Change is a rate the describes how one quantity changes in relation to another Table can be vertical = time is left column Given the table find the rate of change between days 2 and 6? 240 −80 6 −2 = 160 4 = 40 𝑚𝑖𝑙𝑒𝑠 1 ℎ𝑜𝑢𝑟

Tables Find the rate of change between rides 2 and 3. 12.75 −11.00 3 −2 = 1.75 1 = $1.75 1 𝑟𝑖𝑑𝑒 Find the rate of change for days 2 and 3 50.25 −53.92 3 −2 = −$3.67 1 𝑑𝑎𝑦

Rate of Change for a graph = Δ𝑦 Δ𝑥

Quick Quiz x Y 1 9 2 8 3 7 4 6 5 x f(x) = 3x f(x) -2 2 56 2 126 3 176 Function or Not? 2. Fill out the table 3. Find the rate of change between 3 and 5 x Y 1 9 2 8 3 7 4 6 5 x f(x) = 3x f(x) -2 2 Hours Distance in miles 56 2 126 3 176 5 276

7.3 LT: I can find a constant rate of change Define Constant Linear Functioins have a Constant Rate of Change

Check for understanding Does the following have a constant rate of change? Does the following graph have a constant rate of change?

Proportional Relationships – Page 206

Proportional Relationships – Page 206 Example of one that is not Example that is, all ratios are = .45

Everyday use of slope – Handicap accessible The least possible slope shall be used for any ramp. The maximum slope of a ramp in new construction shall be 1:12. The maximum rise for any ramp run shall be 30 inches. (American Disabilities Act) Which means for every foot of rise, you need 12 feet of length.

7. 4 LT I can find the slope between two points 7.4 LT I can find the slope between two points. Slope – steepness of a line

7.4 A lowercase m is used to abbreviate slope Slope = 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛 Scott’s Slope Definition m = 𝑢𝑝 𝑜𝑟 𝑑𝑜𝑤𝑛 𝑙𝑒𝑓𝑡 𝑡𝑜 𝑟𝑖𝑔ℎ𝑡 Given the points (1,1) and (6,3) on a coordinate plane. Can you find the slope? Is it positive or negative?

7.4 Another example

Put in notes - Correct way to find slopes

Examples – remember left to right

Two Special Slopes

Page 484 1 – 22, be careful on 2 and 3 Just by looking, what is the slope ? I feel this is an incorrect way to find slope, Why?

Slope – Day 2 Find the slope between two points ( 𝑥 1 , 𝑦 1 ) ( 𝑥 2 , 𝑦 2 ) (2, 4) (5, 13) Place numbers in formula 13 −4 5 −2 9 3 = 3

( 𝑥 1 , 𝑦 1 ) ( 𝑥 2 , 𝑦 2 ) ( -5, 0) and (-2, -4) −4 −0 −2−(−5) −4 3 ( 𝑥 1 , 𝑦 1 ) ( 𝑥 2 , 𝑦 2 ) (-3, 5) and (-3, 2) 2 −5 −3−(−3) −3 0 Undefined Slope

Slopes – Day 3 Bell Ringer - In your notebook, please complete the following. From your notes on slope; write down at least two ideas about slope in complete sentences, with correct capitalization and spelling. ( 4 minutes )

Example of writing Slope measures the steepness of a line. The slope of a line can be negative, positive, zero, or undefined. The slope formula is the rise of line divided by the run of the line. Slope is always measured from the left point to the right point.

Expanding Slopes to Tables Remember how rate of change was found. Rate of change = 67 −55 5 −1 = 12 4 = 3 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 1 𝑑𝑎𝑦𝑠 It is the same for slope. You will do the same thing except it will be for an x and y table Slope between 1 and 2 = 4−(−5) 2−1 = 9 Date 1 5 8 Temp 55 67 52 x 1 2 3 y -5 4 7

Quick Quiz What is the rate of change between A and B B and C C and F 5 𝑚𝑝ℎ 1 𝑠𝑒𝑐 0 𝑚𝑝ℎ 1 𝑠𝑒𝑐 −2 𝑚𝑝ℎ 1 𝑠𝑒𝑐