FOUNDATIONS OF FUNCTIONS

Cost FOUNDATIONS OF FUNCTIONS MULTIPLE REPRESENTATIONS Verbal Description: A pizza place charges $ for a cheese pizza and an additional $.5 for each topping. Equation: c = +.5t (c = cost and t = toppings) Function Notation: Function notation shows the output (cost) in terms of the input (toppings). c(t) = +.5t Table: The input is on the left and the output is on the right. Toppings Cost (input) (output) 1.5 t C = +.5t Mapping: The input is in the left bubble and the output is in the right bubble. Arrows connect the corresponding pairs. Toppings 1 Ordered Pairs: An ordered pair relates the input (x) and output (y) values in parentheses. (x, y) (,) (1,.5) (, ) Graph: The points are graphed on a coordinate plane. The input values are on the x-axis and the output values are on the y-axis. 9 8 Cost.5 1 5 1 Toppings Domain: The domain is a list of all the input values in a set of braces. D: {, 1,, } Range: The range is a list of all the output values in a set of braces. R: {,.5,, } INDEPENDENT AND DEPENDENT Independent: The independent part of a relationship is the input, the cause, and the if. It corresponds to the x-axis. Dependent: The dependent part of a relationship is the output, the effect, and the then. It corresponds to the y- axis. When identifying independent and dependent, it will be a word or letter NOT a number. To determine the independent and dependent parts of a relationship from a: Table: Independent is on the left and dependent is on the right (for a vertical table) Independent is on the top and dependent is on the bottom (for a horizontal table) Identify the independent and dependent variables. # of days 1 Cost 15 5 Independent: # of days Dependent: Cost Identify the independent and dependent variables. # of miles Time 5 4.5 Independent: # of miles Dependent: Time Equation: Independent is in the math and dependent is by itself. t = 4c + 5 Independent: c Dependent: t h(m) = 15 15m Independent: m Dependent: h(m) -d + = e Independent: d Dependent: e Ordered Pair: Independent is the x value and dependent is the y value. (-, 9) Independent: x (NOT -) Dependent: y (NOT 9) Mapping: Independent is the left bubble and dependent is the right bubble. people cost 8 1 4 8 Independent: people Dependent: cost

To determine the independent and dependent parts of a relationship from a: Graph: Independent is on the x-axis and dependent is on the y-axis. Examples: {(1, 15), (, 1), (5, 14), (1, -)} Not a function (1 is repeated as an x-value.) {(1, ), (,), (, ), (4, ), (5, )} Function (All x-values are different.) x -1 5 y -4 8 Function (All x-values are different.) Independent: x Dependent: y Independent: # of questions Dependent: Verbal Description: Write and equation for the situation if possible. Or, write a depends on, is a function of, if-then, or determines sentence. The dependent part DEPENDS ON the independent part. The dependent part IS A FUNCTION OF the independent part. IF (independent), THEN (dependent). The independent part DETERMINES the dependent part. My total pay, t, is found by adding my base salary of $5 plus % commission on my sales, s. Identify the independent and dependent variables. EQ: t = 5 +.s Function Notation: t(s) = 5 +.s Independent: s, sales Dependent: t, total pay As my car gets older, it is worth less. Identify the independent and dependent variables. Dependency Statement: worth depends on age Function Statement: worth is a function of age Function Notation: w(a) If/Then Statement: If my car get older, it is worth less. Determines Statement: how old my car is determines its worth Independent: age Dependent: worth The grass gets greener as I put more fertilizer on it. Dependency Statement: green depends on fertilizer Function Statement: green is a function of fertilizer Function Notation: g(f) If/Then Statement: If I put fertilizer on the grass, it gets greener. Determines Statement: fertilizer determines green Independent: fertilizer Dependent: green FUNCTION OR NOT? In order for a relationship to be a function, there can only be one output for each input. All of the x-values have to be different. The y-value does NOT matter. A relationship is not a function if there can be more than one output for each input. If even one x-value is repeated, it is not a function. Vertical Line Test If you can draw a vertical line through any part of the graph and it intersects more than once, it is NOT a function. x y 1 8 Not a Function ( is repeated as an x- value.) 1 the graph of a vertical line Not a Function (All the x-values are the same.) the graph of a horizontal line Function (All the x-values are different.) - -4 - Not a Function (does not pass the vertical line test) Function (passes vertical line test) Not a function (does not pass the vertical line test) Not a function (- is a repeated x-value) Function (Each x-value has only one y- value)

Attention Span (Dependent) DOMAIN AND RANGE Domain: all of the possible x-values (input values) Range: all of the possible y-values (output values) DiLiR the RoBoT (for continuous graphs) D Domain R Range i input (x) o output (y) L Left B Bottom i input (x) o output (y) R Right T Top Left <(<) x <(<) Right Bottom <(<) y <(<) Top D: {- < x < 5} or x < 5 R: {- < y < } or y < Your car holds 1 gallons of gas. What would the reasonable domain and range be for the cost to fill up your tank if gas costs $ per gallon? EQ: c = g D: { < g < 1} and R: { < c < 48} (You can t have negative gallons, but you can have decimal amounts, so it is written in inequality form.) SITUATIONAL GRAPHS When interpreting a situation from a graph make sure to explain how the x-values relate to the y-values. Read the labels of the x- and y- axes. Include these labels in your description. If the graph has multiple parts, make sure to explain each section of the graph. When creating a graph for a situation, make sure to label the x-axis with the independent variable and the y-axis with the dependent variable. Include numbers if appropriate. D: {- < x < } R: (- < y < } Speed is increasing at a constant rate. Speed remains constant. D: {- < x < } or x - R: (- < y < } or all real numbers Speed is decreasing at a constant rate. D: {-4 < x < 4} R: {- < y < } Speed increases at a constant rate then suddenly decreases at that same constant rate -1 9 4-1 D: {-1,, } R: {, 4} D: {9} R: {-1, } Joe Sue Bob Joe fastest at first but stopped and then continued at a slower pace Sue constant throughout Bob slowest at first but increased at the end 1 st : Bob nd : Sue rd : Joe You sell candy at a football game for $1.5 each. You paid $5 for candy bars. What is the domain and range for your profit? EQ: p = 1.5c 5 D: {, 1,, } and R: {-5, -48.5, -4, 4} (You can t sell negative or decimal amounts of candy bars, so it has to be written as a list.) Draw a graph to represent the situation: The ideal temperature for a classroom is around. If the temperature rises above that, the student s attention span decreases. Similarly, if the temperature falls below that, the attention span also decreases. Temperature (Independent)

Algebra Grade Temperature Height Savings PARENT FUNCTIONS Linear Parent Function the equation that all other linear equations are based upon (y = x) Quadratic Parent Function the equation that all other quadratic equations are based upon (y = x ) PROPORTIONAL OR NON-PROPORTIONAL A situation is proportional if increases or decreases at a constant rate without an initial amount or anything added (or subtracted) from it. You get paid $.5 an hour at Kroger constant rate where hours gets you $ A waiter gets paid $1 an hour plus tips Not constant rate but it has an additional amount added to it Absolute Value Parent Function the equation that all other absolute value equations are based upon (y = x ) A boy rode his bike miles home Not This is a constant distance not a rate! CORRELATION Positive Correlation a set of data that has an OVERALL trend that goes up (positive slope) EXAMPLE: The amount of money you have if you save some each week. A linear function is proportional if the line goes through the point (, ). A linear function is not proportional if the line does NOT go through (, ). Weeks Negative Correlation a set of data that has an OVERALL trend that goes down (negative slope) EXAMPLE: The height of a candle as it burns Not Linear but NOT through (,) Linear AND through (,) Time A table of values is proportional if y divided by x is constant and it has a function rule that does NOT add or subtract anything. (To find the rate divide the change in y by the change in x for each pair of points.) Constant Correlation a set of data that relatively remains the same (zero slope) EXAMPLE: the temperature of a room if the thermostat is set on 5 Time or Not? x y - -1 y/x = 8-1 -8 y/x = 8 4 y/x = 8 48 y/x = 8 Function Rule: y = 8x y divided by x is constant (8) and nothing is added or subtracted. No Correlation a set of data that does not appear to have any relationship EXAMPLE: shoe size and Algebra grade or Not? x -5-1 5 y 1-5 -1 y/x = y/x = y/x = y/x = -1.4 -.5-5 -. Not y divided by x is NOT constant. Shoe Size

PATTERN BUILDING (FUNCTION RULES) Linear data that would make a straight line if graphed Zero Term the initial value, original amount or constant Rate of Change the amount a pattern increases or decreases each Function Rule an equation that can be used to fine values in a pattern Term # # Blocks 1 Zero Term 1 + 5 Rate of Change + n 1 + n Function Rule LINEAR OR NOT To see if a table is linear, find the change in y divided by the y change in x ( )between each consecutive data points. If x y the is the same, the data is linear. Linear does NOT have x to be proportional. It can have something added or subtracted. EXAMPLE: Linear or Not? -5-(-) = - - - 1 = - 1-4 = - x -5-1 4 y 1-5 -11 TI84 - CALCULATOR (GRAPH POINTS AND CHECK FUNCTION RULE) Enter Data: 1. Edit;. Enter independent (x) values in and dependent (y) values in y x 1 = 1 (-5) = -5 (-11) = ) is the same for all points ( y x Clear Data: 1. Edit;. to highlight (or );. Graph Points: 1. (for );. on Plot 1;. Highlight and turn plot on; 4. Turn off Stat Plot: 1. (for );. ENTER on Plot 1;. Highlight and turn plot off Change Window: X min = smallest x X max = biggest x X scl = count by # Y min = smallest x Y max = biggest x Y scl = count by # Original Window: 1. Graph a Function Rule: 1. button for the variable) ;. :ZStandard ;. Type the rule in y 1 (use the Check a Function Rule: 1. Enter the function rule into ;. (to check the ) Clear Function Rule: 1. ;. To see if a set of ordered pairs or mapping is linear, graph the points to see if it makes a straight line. Is (, -1), (, ), and (-, 4) linear? No. It would not make a straight line. Yes. It graphs a straight line. To see if a function rule (equation) is linear, type the equation into Y= and Graph. Is y = x 5 linear? Yes Is y = -x + linear? No To see if a verbal description is linear, write an equation and graph it or check to see if the rate of change is constant. A car at a red light accelerates after the light turns green. No. The rate of change is increasing. -4-5 - You spend $4 every day for lunch. Yes. The rate is constant.