Function vs Relation Notation: A relationis a rule which takes an inputand may/or may not produce multiple outputs. Relation: y = 2x + 1

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1 /0/07 X Y Relations Vs. Functions Warm up: Make a table of values, then graph the relation + Definition: A relationis a rule which takes an inputand ma/or ma not produce multiple outputs. A functionis a special relation where ever input has at most one output. Function vs Relation Notation: Relation: + Read it f of : the brackets here DO NOT MEAN MULTIPLY!! Function: f() + E : find f(7) (7) + find f(-) (-) + -9 Consider machines. Machine A: Buttons Push INto order A FUNCTIONING MACHINE Drink Comes OUT Vertical line test (VLT) Drink Button Machine B: Buttons Push INto order Drink Comes OUT A Not FUNCTIONING MACHINE Drink Vertical line test (VLT) Button () Mapping Diagram Function YES 7 A Source Input Independent X-values Antecedent Domain f B 0 Target Output Dependent Y-values Image Range Each source value has onl ONE target value Function NO 7 A r B 0 A source value has MORE THAN ONE target value () Table of Values Function YES 0 8 Each value has onl ONE value. Function NO 0 An value has MORE THAN ONE value () Graph - Vertical Line Test (VLT) Function YES Function NO E : Evaluate the following functions a) f() + find f() f() () + b) g() + find g() g() () + 7 Pages 9,9 # - c) h() (-)(-) find h() A vertical line touches the curve at ONLY spot at a time A vertical line touches MORE THAN spot at a time. 7 h() (-)(-) ()() 8 9

2 /0/0. A- Modes of Representation of a function In a relation between variables and, one usuall depends on the other (the output depends on the input). We sa: depends on therefore is the dependent variable, and is the independent variable. Do activities,, on pages 9, 97 P. 9 Act. : A ferr ensures the transportation from a town to an island. The rates are the following: $0 per car and $0 per occupant in the car. No car is accepted without an occupant and there is a maimum of occupants per car. a). : # of occupants. : cost b) Because ever -value in set A has at most one image (-value) in set B c) ) f() ) 0; it costs $0 for a car with occupants ) ; it costs $70 for a car with occupants d) Y increases too. A f B P.9 Act.: Anna works as a dental hgienist in a clinic. Her hourl wage is $. In this situation consider the following two variables: the number of hours worked in a week and her salar. a) b) Yes c). # hours of work. Salar in $ d) f() e) it increases # hours worked Salar($) P. 97 Act : A water reservoir contains 000 liters of water. A pump is activated to empt the reservoir at a rate of 0 liters per minute. Consider the function which associates the variable elapsed time with the variable quantit of water left in the reservoir. a). Elapsed time in min. Quantit of water left b) Time (min) Water left (l) c) d) q() 000 0t e) it decreases Page 98 #

3 /0/0.-B- Modes of Representation of a function There are different was of representing a function:. Verbal/Written. Rule/Equation That is a sentence/paragraph to describe the. Table of values function in words.. Cartesian graph or Mapping diagram. Verbal/Written: E: A repairman charges $0 per hour plus $0 for his travel epenses.. Rule/Equation: That translates from English to Math, and epresses the dependent variable in terms of the independent variable. E: : hours (indep. Var.) : cost in $ (dep. Var.) or C() Table of Values: A wa to organize data. It associates the values with their values. E: Hours 0 Cost Cartesian graph or Mapping Diagram: A visual Representation E: Cost of Repair Page 99 # - Page 0 #,7 Remarks:. The indep. Var. goes on the Horizontal ais; the dep. Var. goes on the vertical ais.. Choose an appropriate scale for the and -ais.. We can break the ais if the graph starts up too high.. Remember to label the ais, the scales, and put a title.

4 . -A-Rate of change Rate of change(r.o.c) is also known as Rate of variation, Slope, and Rise over Run Given points A(, ) & B(, ) on a line, then the rate of change between points A & B is : R.O.C. ; : difference or change in E : R.O.C. Given points A(0,) and B(,) Or Count or if ou have a graph On an straight line all segments of the line will have the same slope possible slopes are noted:. Positive slope :. Negative slope: E :Find the slope of each line on the grid: Slope of line: a: b: horizontal line so slope 0. Zero slope:. Undefined slope: c: d: vertical line so slope is undefined E : Find the slope of the following: ) A(,) B(,) ) C(-,) D(,8) rise run - - () - () () - () - rise run - - (8) - () () - (-) page 0 # - -

5 /0/0. -B- Constant Rate of Change Find the rate of changes: a) Between A and D: R.O.C b) Between D and C: R.O.C c) Between C and A: d) Between A and B: R.O.C R.O.C Propert of ROC of a straight line: On a straight line the ROC is constant ( the same) no matter which points we pick. Where as on a curve the ROC changes (is not constant) along the curve. On a graph we can count the units: ROC Given points A(, ) and B(, ): ROC Find the rate of Change for the Following Find the rate of Change for the Following E : A tank initiall contains 000 L of eggnog. The tank is leaking! After hrs the tank has 000 L. The points would be: A (0, 000) and B (, 000) R.O.C. change in change in - - (000 L)-(000 L) -00 L/hr 0 hrs - hrs E : Jen earns $0 for 0 hours of work. For 0 hours of work she earns $0. What is her hourl rate? R.O.C. change in change in - - ($0)-($0) R.O.C. 0 hrs - 0 hrs R.O.C. $/hr Hours Pa E : The slope is G(0,-) and B(c,) a - - ()- (-) (c) (0) 8 c c Find the value of c E : The slope is / S(-,) and T(c,) a c c ()- () (c) (-) c+ c 7 page 0 # -7

6 /0/0. Polnomial Functions Recall: a Polnomial is a monomial, or a sum/difference of monomials, with variables whose eponents are onl natural. Warm up: Determine the degree of each function. f() + f() 0 f() - + f() + f() -+ f() none A polnomial functionis an function whose rule is a polnomial. Degree Tpe of function 0 Constant Linear Quadratic Direct Linear function Partial Linear function Properties:. -A- Constant Functions. The ROC 0 ( aka: zero variation function). The rule: b or f() b. Table of values:. Graph: Horizontal line passing through the -ais at b E : The movie ticket costs $9 for all ages. If is the age and is the cost of ticket Table of values: ROC: The rule is : The graph: 0 9 X Y Now tr activit Page 07

7 /0/0. -B- Linear Functions Linear Functions have degree The rule: or a + b f() a + b Case : if b 0 Direct Variation Linear function Case : if b 0 Partial Variation Linear function Case : if b0 Direct Variation Linear function Properties:. Ever -value is a direct multiple of the -value. The rule: a or f() a. Table of values: Graph: a Diagonal line passing through origin (0,0) Case : if b 0 Partial Variation Linear function Properties:. -values are not direct multiples of the -values. The rule: a + b or f() a + b. Table of values: Graph: a Diagonal line passing through -ais at point (0,b) -intercept E : is it Constant, Direct Linear, Partial Linear or Other O C O E : is it Constant, Direct Linear, Partial Linear or Other 7 9 P.L D.L P.L. E : is it Constant, Direct Linear, Partial Linear or Other + P.L. f() f() O O D.L. C P.L.

8 /0/0 Da : Page # - Tips for graphing a linear function: ) Make a Table of Values (min points) ) Choose eas -values like. 0, or If our slope is a fraction, pick multiples of the value for the run (denominator). - ie if slope / pick 0,, Graph and label these equations Da : Page # -7 Da : Page # 8-0

9 /0/07. Finding the Rule of the Linear function A straight line alwas follows the RULE a + b Where: is the Dependent variable is the Independent variable a is the R.O.C.(slope) b is the initial value (-intercept) Steps to Finding the RULE given points Step : Find the slope using a - - Step : Find the -intercept (b) b plugging the (, ) coordinates into b a Step : State the final equation. a + b E: Find the rule of the line going through (-,) & (-, ) Step : Find a a - - () - () (-) - (-) a Step : Final equation Step : Find b using (-,) b a b() - (-) b() + b E : Find the rule of the line going through (-,) & (, ) Step : Find a a - - () - () () - (-) a - a - Step : Final equation Step : Find b using (,) b a b () - (-)() b () + b - + A table of values shows the relationship between two variables, tpicall and. time height 0.. We call the independent variable because we choose it. is the dependant variable because it depends on the chosen value of. TOV Tpe - Both a and b are clear Run Rise a - b TOV Tpe - a is clear, must find b Run Rise Find b Sub (,7) and a in b - a b a b (7) - ()() b 7 b a b + 7 TOV Tpe - a not clear and must find b Run Rise - -8 Find b Sub (,) and a in b a b a b () - ()() b 0 Rise - b - -8 Run a b Page 0 # -7 9

10 0//0. Sstems of st Degree Equations Finding the solution to a sstem of equations means find a common point (,), that fits into both equations at the same time. We can find the solution b making a table of valuesand finding when the values for are the same. We can check the solution of a sstem b replacing it back into the original equations to see if it works. We can also graphthe two lines on the same grid and see where the lines cross. E : Is and a solution to the following sstems? Works for the first equation but not for the second. Therefore (,) is not a solution. Works for both equations. Therefore (,) is a solution. E : Solve the sstem using a table of values Choose values for and calculate values for. The solution eists when both values of are the same. solution is (,) Check: + ( ) E : Solve the sstem in e, b graphing E : Solve using the comparison method + - Both equations must be isolated for the same variable E :Solve using the comparison method ( ) + Solve for, Compare both s Replace and solve for then in to check ( ) solution is (-,-) ( ) ( ) + page #, page # -7 7

11 /0/07.7 Parameters a and b in a + b For ever line a + b ( the parameters are a and b, and the each affect the look of the line) We can observe the affects of changing the parameters b using technolog: TI-8 or GeoGebra or graphsketch.com A 0. tpe of function Direct Direct Direct R.O.C. (a) 0. Initial value (b) Description of line Centered in st and rd quadrants. Increasing line. Steeper than. Increases faster. Bigger angle of Less steep than. Increases slower. Smaller angle of B - Y tpe of function Direct Direct Direct R.O.C. (a) Initial value (b) Description of line Centered in nd and th quadrants. Reflection of. Decreasing line. Steeper than. decreases faster Bigger angle of Less steep than. decreases slower. Smaller angle of C tpe of function Direct Partial Partial R.O.C. (a) Initial value (b) 0 - Description of line Centered in st and rd quadrants. Increasing line. Parallel to Translated (shifted) up units Parallel to Translated (shifted) down units D tpe of function Partial Partial Partial R.O.C. (a) 0. - Initial value (b) - Description of line times steeper than Shifted up units Half as steep as Shifted down units times steeper than and reflected Shifted up units Conclusions: For ever line a + b ( the parameters are a and b affect the look of the line) a : affects the angle of inclination (steepness of line) b : affects the vertical translation of the line

12 .7 Parameters a and b in a + b Observations on changing the parameters using: TI-8 or GeoGebra or graphsketch.com A 0. tpe of function Direct Direct Direct R.O.C. (a) 0. Initial value (b) Description of line Centered in st and rd quadrants. Increasing line. Steeper than. Increases faster. Bigger angle of Less steep than. Increases slower. Smaller angle of B - Y tpe of function Direct Direct Direct R.O.C. (a) Initial value (b) Description of line Centered in nd and th quadrants. Reflection of. Decreasing line. Steeper than. decreases faster Bigger angle of Less steep than. decreases slower. Smaller angle of C tpe of function Direct Partial Partial R.O.C. (a) Initial value (b) 0 - Description of Centered in st and Parallel to line quadrants. Translated (shifted) Increasing line. up units D tpe of function Partial Partial Partial R.O.C. (a) 0. - Initial value (b) - Description of line Parallel to Translated (shifted) down units times steeper than Shifted up units Half as steeper as Shifted down units times steeper than and reflected Shifted up units Conclusions: For ever line a + b ( the parameters are a and b affect the look of the line) a : affects the angle of inclination (steepness of line) b : affects the vertical translation of the line

13 //0.8 Rational function Act. Page : Savannah wants to repaint the offices at work. The job requires 0 hours of work for one emploee. In this situation, consider the function f which associates the number of emploees hired for the job with the time that it takes to complete the job. a # emploees Duration (in hours) b) is not a multiple of, so and are notdirectl proportional. c) ; so R.O.C is not constant d) as increases the decreases, so and don t var in the same direction. e) The productof and is constant( alwas 0). 0 f) The rule is.8 Rational function Act. Page : Savannah wants to repaint the offices at work. The job requires 0 hours of work for one emploee. In this situation, consider the function f which associates the number of emploees hired for the job with the time that it takes to complete the job. # emploees Duration (in hours) g) The graph is a curve that approaches, but never touches either ais. In a Rational Function Variables and are inversel proportional That is as increases decreases. Page # -7 The rate of change is not constant. The product of each pair and is constant. The rule is: k because k The graph looks like

14 /0/07 Compare If Age Height Marks + Stamp 0 collection Game score.9 Inverse of a function Imagine friends: Then asmin avier Sometimes we need to inverse a function to epress it in terms of the other variable. Eamples: ) If P Then ) If A 00 m Then h And b ) If C 0 + n Then n Area 00m b h. In a Direct Variation situation. In a Rational Variation situation. In a Partial Variation situation If a then If a + b then If then To find the inverse of a function: Swap the s and s of each co-ordinate. E. : Function A f() Inverse of A f - () + /-. E : Graph and its inverse Function A Inverse of A Page 0 #,, /+