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IDENTIFYING & REPRESENTING FUNCTIONS Essential Question? How can you identify & represent functions? 8.F.1 COMMON CORE STANDARD: 8.F.1 ─ Define, evaluate, and compare functions. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output OBJECTIVES: • Understand that a function is a rule that assigns to each input exactly one output. • Identify whether a relationship is a function from a diagram, table of values, graph, or equation. Curriculum Vocabulary Function (función): A relationship between an independent variable, x, and a dependent variable, y, where each value of x (input) has one and only one value of y (output). Relation (relación): Any set of ordered pairs. Input (entrada): A number or value that is entered. Output (salida): The number or value that comes out from a process. Curriculum Vocabulary Domain (dominio): The set of all possible input (x) values. Range (rango): The set of all output (y) values. Continuous graph (gráfica continua): A graph of points that are connected by a line or smooth curve on the graph. There are no breaks. Discrete graph (gráfica discreta): A graph of isolated points. Curriculum Vocabulary Linear Graph (gráfico lineal): A graph that is a line or a series of collinear points. Collinear Points (puntos colineales): Points that lie in the same straight line. Non-linear graph (gráfico no lineal): A graph that is not a line and therefore not a series of collinear points. Set (grupo): A collection of numbers, geometric figures, letters, or other objects that have some characteristic in common. REPRESENTING FUNCTIONS There are 4 (FOUR) ways to represent a function that we will explore: 1. TABLE 2. MAPPING DIAGRAM 3. EQUATION 4. GRAPH FUNCTIONS The diagram below shows the function “add 2.” Input = 3 Function: Add 2 Output = 5 There is only one possible output for each input. The function “add 2” is expressed in words. It can also be: • • • • written as the equation y=x+2 represented by a table of values represented as a mapping diagram shown as a graph. IDENTIFYING FUNCTIONS Look at the following table: INPUT OUTPUT 5 11 10 21 15 31 20 41 25 51 For EACH INPUT THERE IS EXACTLY ONE OUTPUT. You can notice that there is NO REPETITION in the INPUT column. This table represents a function. IDENTIFYING FUNCTIONS Look at the following table: INPUT OUTPUT 3 9 3 10 5 25 5 26 7 49 For EACH INPUT THERE IS MORE THAN ONE OUTPUT. You can notice that there is REPETITION in the INPUT column. This table DOES NOT represent a function. IDENTIFYING FUNCTIONS Let’s examine the following situation: Carlos wants to buy some apps for his smartphone. Zynga is offering a game app special. 2 apps will cost him $2.58. 5 apps will cost him $6.45. Help Carlos complete the table. Number of Apps Rule Total Cost 2 5 1 x There is a ONE to ONE relationship! This represents a FUNCTION! IDENTIFYING FUNCTIONS Let’s use the data we found to create a MAPPING DIAGRAM. Number of Apps (x) Total Cost in $ (y) 2 2.58 5 6.45 1 1.29 x 1.29x Input: Number of Apps Output: Total Cost in $ 1 1.29 2 2.58 5 6.45 There is a ONE to ONE relationship! This represents a FUNCTION! IDENTIFYING FUNCTIONS Does the following mapping diagram represent a function? 5 7 -3 2 10 9 -15 4 15 11 -21 6 20 45 -121 8 25 IDENTIFYING FUNCTIONS Does the following mapping diagram represent a function? 1 10 2 20 3 11 5 30 4 40 5 50 IDENTIFYING FUNCTIONS The third way we can represent a function is by writing an EQUATION. In the eighth grade, recognizing if an EQUATION is a FUNCTION is super easy. If you can get y all alone on one side of the equal sign, it is a function! Examples: 𝒚 = 𝟏𝟐𝒙 + 𝟓 𝒚 = 𝟑𝒙𝟐 − 𝒙 + 𝟗 11𝒚 − 𝟒𝒙 =7 IDENTIFYING FUNCTIONS So far we have seen a function represented as: • A TABLE • A MAPPING DIAGRAM • An EQUATION Input 1 2 3 Output 3 6 9 4 12 5 15 1 3 2 6 3 9 4 12 5 15 𝑦 = 3𝑥 IDENTIFYING FUNCTIONS The fourth way to represent a FUNCTION is as A GRAPH: For A GRAPH to represent a FUNCTION, it must pass the VERTICAL LINE TEST. Pass a vertical line over the entire graph. If at any time it touches more than one point at the same, it is IDENTIFYING FUNCTIONS Is this a function? Continuous or discreet? IDENTIFYING FUNCTIONS Is this a function? Continuous or discreet? IDENTIFYING FUNCTIONS Is this a function? Continuous or discreet? IDENTIFYING FUNCTIONS Is this a function? Continuous or discreet? IDENTIFYING FUNCTIONS Is this a function? Continuous or discreet? IDENTIFYING FUNCTIONS We have now seen a FUNCTIONS represented as: • A TABLE • No repetition in the input (x-values) • A MAPPING DIAGRAM • Shows a ONE to ONE relationship • An EQUATION • You can get y all alone on one side of the equal sign • A GRAPH • Passes the vertical line test Input Output 1 3 1 3 2 6 2 6 3 9 3 9 4 12 4 12 5 15 5 15 𝑦 = 3𝑥 FUNCTIONS For EACH INPUT THERE IS EXACTLY (ONE AND ONLY) ONE OUTPUT. IDENTIFYING DOMAIN & RANGE From a TABLE: INPUT OUTPUT 5 11 10 21 15 31 20 41 25 51 DOMAIN: list the x values {5, 10, 15, 20, 25} RANGE: list the y values {11, 21, 31, 41, 51} IDENTIFYING DOMAIN & RANGE Identify the domain and range: INPUT OUTPUT 2 7 3 7 4 7 5 7 6 7 DOMAIN: list the x values {2, 3, 4, 5, 6} RANGE: list the y values {7} IDENTIFYING DOMAIN & RANGE From a GRAPH: First IDENTIFY all the ORDERED PAIRS (-10, 4), (-5, 4), (-4, -6), (-3, 8), (3, 2), (3, -3), (6, 9), (8, 3), (8, -5) DOMAIN: list the x values {-10, -5, -4, -3, 3, 6, 8} RANGE: list the y values {4, -6, 8, 2, -3, 9, 3, -5} {-6, -5, -3, 2, 3, 4, 8, 9} Is the graph continuous or discreet? IDENTIFYING DOMAIN & RANGE Identify the domain and range: Is the graph continuous or discreet? FOR ACCELERATE CLASSES ONLY Objective: To identify dependent & independent quantities To identify the domain and range of a function To recognize, evaluate, and express functions using function notation. Curriculum Vocabulary Dependent quantity (cantidad de dependientes): When one quantity depends no another in a problem situation, it is said to be the dependent quantity. Independent quantity (cantidad independiente): The quantity that the dependent quantity depends upon is called the independent quantity. Circle the independent quantity and underline the dependent quantity in each statement: • the number of hours worked and the money earned. • your grade on a test and the number of hours you studied. • the number of people working on a particular job and the time it takes to complete a job. • the number of games played and the number of points scored. • the speed of a car and how far the driver pushes down on the gas pedal. IDENTIFYING DOMAIN & RANGE From a GRAPH: Is the graph continuous or discreet? Continuous Functions Most CONTINUOUS FUNCTIONS are shown on the graph by using arrows. It means that the x-values of the function continue off to infinity in both directions. DOMAIN: The domain of a continuous function with arrows in both directions will always be ALL REAL NUMBERS. RANGE: The range of a continuous function will vary. In this case the y-values also go off to infinity, so the domain is also ALL REAL NUMBERS. IDENTIFYING DOMAIN & RANGE From a GRAPH: Continuous Functions DOMAIN: ALL REAL NUMBERS. RANGE: In this function what is the y-value? You can see that the y-value will ALWAYS be 6. {6} IDENTIFYING DOMAIN & RANGE From a GRAPH: Non-functions DOMAIN: In this graph the x-value will always be -2. {-2} RANGE: Here the y-values go off to infinity in both directions. ALL REAL NUMBERS IDENTIFYING DOMAIN & RANGE From a GRAPH: Non-functions DOMAIN: For geometric shapes and line segments, you need to determine where the x-values are trapped. In this example, the triangle is trapped between -5 and 9. -5≤x≤9 RANGE: For geometric shapes and line segments, you need to determine where the y-values are trapped. In this example, the triangle is trapped between 1 and 8. 1≤y≤8 IDENTIFYING DOMAIN & RANGE From a GRAPH: What about this one? DOMAIN: Since it is a continuous function the domain is: ALL REAL NUMBERS. RANGE: This functions lowest value is -6. Then it goes up from there to positive infinity. y≥-6 FUNCTION FORM FUNCTION FORM means to get y all alone on one side of the equal sign. y stuff FUNCTION NOTATION Function Notation is another name for the letter y. The same way a person whose name is José, might be known as Pepe, function notation is another name for the same thing. Function notation looks like this: y f (x) We say y equals f of x. This does NOT mean multiply! FUNCTION NOTATION y f (x) In function form, the variable that is all alone, y, is the DEPENDENT VARIABLE. The variable in the parenthesis, x, is the INDEPENDENT VARIABLE. The value of y depends on what you plug in for x. Function Notation y 2x 3 f (x) 2x 3 when x 1, y 5 f (1) 5 when x 2, y 7 f (2) 7 when x 3, y 9 f (3) 9 when x 4, y 11 f (4) 11 f ( 4) 5 g(x) x 2 h(x) 3x 2 Evaluate the following. 1) g(4) 16 5) h(4) g(1) 2) h( 2) 8 6) h( 5) g( 2) 3) g( 3) 9 4) h(5) 13 10 1 11 17 4 68 7) g h(3) g(7) 49 8) h g(2) h(4) 10 Evaluate the function over the domain, x = -1, x = 0, x = 2. 1) f (x) 4x {4, 0, 8 } 2) g(x) 3x 9 {12, 9, 3 } 3) h(x) x 1 2 { 0, 1, 3 }