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3 ESO BILIES MANUEL CAADASMATHS NFC(Source: Fina Cano and Boardworks)FUNCTIONS1T.6 FUNCTIONS AND GRAPHSIntroductionFunctions and their graphsVariations in a functionMaxima and minimaTrends and periodicityContinuity and discontinuity2IntroductionA little of history:La gripe espaola

Between 1918 and 1919 the Spanish flu (H1N1) killed 50 million people all over the world, 300.000 in Spain, 650.000 in USA. It was much worse than the black death of the XIV century.4

A little of history:La gripe espaola5

A little of history:La gripe espaolaModeling the real world7Modeling the real worldIn every day life, many quantities depend on one or more changing variables. For example:Plant growth depends on sunlight and rainfall.Distance travelled depends on speed and time taken.Test marks depend on attittude, attention, study (among many others variables!).

8Modeling the real worldIn this unit we are going to study functions. Functions are very useful to model real situations where one quantitydependson another quantity.Graphs are a good way of presenting a function, they give us a visual picture of the function.

9Modeling the real worldDans journey on his bike.

Dan start his journey covering 10 miles in 1 hour. Then he stops for 1 hour to rest. Forthe following half an hour he goes faster because he goes downhill. After that he stops for one hour and a half to visit a friend. Finally, he returns home covering 25 miles in 3 hours.

10Modeling the real world

This graph shows Dans journey on his bike. 11Filling flasks 1

12Start by explaining that the we are going to produce a graph of the depth of water in a flask as it fills with water. Note that the water flows out of the tap at a constant rate.As the first flask fills up the graph of depth against time will be drawn. Ask pupils to tell you how many cm are filled each second for the flask.Ask pupils to predict the slope of the graph for the new flask compared to the previous flask. Ask pupils to justify why they think the graph will be steeper or less steep than before.Continue for each flask in turn. Establish the depth in the narrowest flask will increase the fastest and therefore produce the steepest graph. The depth in the widest flask will increase the slowest.Ask pupils to explain why all the lines pass through the origin. Ask pupils to explain why the lines are straight.Ask pupils to explain what would happen if the water from the tap did not flow out at a constant rate. For example, in real life the rate of the water coming out of the tap would speed up as the tap is turned on. How would this affect the shape of the graph?Filling flasks 2

13Start by explaining that the we are going to produce a graph of the depth of water in a flask as it fills with water. Note that the water flows out of the tap at a constant rate.As the first flask fills up the graph of depth against time will be drawn.Ask pupils to predict the shape of the next graph before it is drawn, justifying their explanations. Establish that the wider the flask is at a given point the loner it will take for the water to increase in depth.Continue for each flask in turn. Ask pupils to explain why all the lines pass through the origin. Ask pupils to explain what would happen if the water from the tap did not flow out at a constant rate. For example, in real life the rate of the water coming out of the tap would speed up as the tap is turned on. How would this affect the shape of the graph?Ask pupils if we can use the graph to work out the capacity of each flask.Modeling the real worldThis graph models the depth of the water flowing in or out of a container at a constant rate.

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Matching graphs to statements

15Establish that if something is rising rapidly over time it will have a steeper and steeper gradient. If something is falling rapidly it will have a steeper negative (or downward) gradient. If something is rising or falling steadily then the graph will be straight.Ask pupils what a horizontal section of graph would represent.Each of the graphs in this example illustrates trends rather than accurate information.Functions and their graphsWhats a function?is a function but is not.17Function is a relation between two variables such that for every value of the first (usually x), there is only one corresponding value of the second (usually y). Whats a function?is a function but is not.18Function is a relation between two variables such that for every value of the first (usually x), there is only one corresponding value of the second (usually y). Whats a function?is a function but is not.

19 Not all the graphs are functions! Function or not?is a function but is not.

20 Not all the graphs are functions! Function or not?is a function but is not.

21 Not all the graphs are functions! Function or not?is a function but is not.

22Whats a function?is a function but is not.We say that the second variable is a function of the first variable.Lots of functions can be given by formulas, called analytical expressions of the functions.23Whats a function?is a function but is not.

We say that the second variable is a function of the first variable.Lots of functions can be given by formulas, called analytical expressions of the functions.We normally write the function as f(x), and read this as function of x (y and f(x) are the same).

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Using an equationA function is a rule which maps one number, sometimes called the input or x, onto another number, sometimes called the output or y.A function can be illustrated using a function diagram to show the operations performed on the input.For example: A function can be written as an equation. For example, y = 3x + 2.A function can can also be be written with a mapping arrow. For example, x 3x + 2.xy 3+ 22525Ask pupils if they know what a function is and if they know the difference between a formula and a function (these two words are sometimes confused).Formulae and functions can both be written as equations using an equals sign. Functions can also be written using a mapping arrow, for example, x 3x + 2.A function is a rule which maps one number, sometimes called the input, onto another number, sometimes called the output, using a sequence of operations. A formula, on the other hand, can link more than two quantities, for example A = lw. There does not need to be a unique output for a given input. Also, formulae relate to real-life or practical contexts while functions usually relate to abstract inputs, outputs and operations.Discuss the function y = 3x + 2 in terms of inputs and outputs. The function y = 3x + 2 means that the input, x, is multiplied by 3 and then 2 is added on to get the output, y.Ask pupils, to find y for different values of x. Then ask,If y is 38, what is x? Explain that if we are given the output and asked to find the input then we have to use inverse operations. Ask, What is the inverse of times 3 and add 2? Tell pupils that we must not only invert the operations but we must also reverse the order. Establish that the inverse of times 3 and add 2 is, subtract 2 and divide by 3. Performing these inverse operation in reverse order, we have x = 12 when y = 38 (because (38 2) 3 = 12).

Using an equation: The function machineIs there any difference betweenxy 2+ 1andThe first function can be written as y = 2x + 1.The second function can be written as y = 2(x + 1) or 2x + 2.xy+ 1 2?2626Show that these machines are different by using the same input in both machines and comparing the outputs.

Finding outputs given inputs

2727Start with one machine to demonstrate to pupils how the function machine works. Explain that a number is fed into the machine (this is the input), the machine performs an operation (or series of operations) on the number to produce the output.Change the number of machines and use the function editor to create a given chain of functions. Hide the output before pressing the orange button. Change the input and ask pupils to find the output.If required you can show the intermediate steps.

Using a tableWe can use a table to record the inputs and outputs of a function.For example,We can show the function y = 2x + 5 asxy 2+ 5and the corresponding table asxy33311113, 111111, 7173, 1, 6 7611, 7, 176173, 1, 6, 417411, 7, 17, 134133, 1, 6, 4, 1.5131.511, 7, 17, 13, 81.582828

Using a table with ordered valuesIt is often useful to enter inputs into a table in numerical order.For example,We can show the function y = 3(x + 1) asxy+ 1 3and the corresponding table asxy111661, 2626, 9291, 2, 3936, 9, 123121, 2, 3, 41246, 9, 12, 154151, 2, 3, 4, 51556, 9, 12, 15, 18518When the inputs are orderedthe outputs form a sequence.2929Stress that when the inputs of a function are entered in numerical order, the outputs form a sequence.Point out that the nth term of this sequence is 3(n + 1) (or 3n + 3). This can be useful when we are given inputs and outputs and need to find the function.Although there are similarities between sequences and functions, there is a crucial difference between n, the term number in a sequence and x, the input of a function. That is that n can only be a whole positive number, whereas x can be any number, positive, negative or fractional.

Recording inputs and outputs in a table

3030Use this activity to demonstrate how functions can be entered into a table.Ask pupils, How could we show this function as a graph?Establish that each value we have chosen for x and the corresponding value for y, form a coordinate pair. Plotting these coordinates would give us a graph of the function.

Drawing a function: Coordinate pairs

When we write a coordinate, for example,Together, the x-coordinate and the y-coordinate are called a coordinate pair.the first number is called the x-coordinate and the second number is called the y-coordinate.(3, 5)x-coordinate(3, 5)y-coordinate(3, 5)the first number is called the x-coordinate and the second number is called the y-coordinate.3131Link:S4 Coordinates and transformations 1 coordinates.

Drawing graphs of functions

The x-coordinate and the y-coordinate in a coordinate pair can be linked by a function.What do these coordinate pairs have in common?

(1, 3), (4, 6), (2, 0), (0, 2), (1, 1) and (3.5, 5.5)?In each pair, the y-coordinate is 2 more than the x-coordinate.These coordinates are linked by the function: y = x + 2We can draw a graph of the function y = x + 2 by plotting points that obey this function. 3232Ask pupils if they can visualize the shape that the graph will have. This might be easier if they consider the points (0, 2), (1, 3), (2, 4) (3, 5) etc. Establish that the points will lie on a straight diagonal line. Stress that the graphs of all linear functions are straight lines. A function is linear if the variables are not raised to any power (other than 1). Ask pupils to suggest the coordinates of any other points that will lie on this line. Praise the most imaginative answers.

Drawing graphs of functions

Given a function, we can find coordinate points that obey the function by constructing a table of values.Suppose we want to plot points that obey the functiony = x + 3We can use a table as follows:xy = x + 332101230(3, 0)123456(2, 1)(1, 2)(0, 3)(1, 4)(2, 5)(3, 6)3333Explain that when we construct a table of values, the value of y depends on the value of x. That means that we choose the values for x and substitute them into the equation to get the corresponding value for y.The minimum number of points needed to draw a straight line is two, however, it is best to plot several points to ensure that no mistakes have been made.The points given by the table can then be plotted to give the graph of the required function.

Drawing graphs of functionsto draw a graph of y = x 2:1) Complete a table of values:2) Plot the points on a coordinate grid.3) Draw a line through the points.4) Label the line.5) Check that other points on the line fit the rule.xy = x 23210123For example,yxy = x 254321013434This slide summarizes the steps required to plot a graph using a table of values.

Drawing graphs of functions

3535Start by choosing a simple function.Remind pupils that we can draw graphs of functions by plotting inputs along the x-axis against outputs along the y-axis.Talk through the substitution of each value of x in the table and click to reveal the corresponding value of y below it. Start with the positive x-values, if required and work backwards along the table to include the negative values.Explain that each pair of values for x and y corresponds to a coordinate that we can plot on the coordinate axis. For example, for the equation y = 2x, when x = 1, y = 2. This corresponds to the coordinate (1, 2). Click to plot each coordinate from the table of values onto the graph. Remind pupils that we always move along the x-axis and then up (or down) the y-axis when plotting coordinate points. A common mnemonic for this is along the corridor and up the stairs.Once all the points have been plotted ask pupils what they notice, that is that all the points lie in a straight line.Click show line to draw a line through the points. Draw pupils attention to the fact that the line extends beyond either end of the points plotted on the graph. Use the crosshair button to find the coordinates of other points on the line. Verify that all of these points satisfy the equation. Ask pupils to suggest other coordinates that would lie on this line. Establish that the line could be infinitely long and praise the most imaginative correct answers. Not all the analytical expressions are functions! Function or not?is a function but is not.

36 Not all the analytical expressions are functions! Function or not?is a function but is not.

37 Whats a function?As a conclusion:A Function is a relation between two variables, x and y, which associates each value of x to a single value of y.This relation can be expressed using:Equation (mathematical relationship)Table (pair of values for x and y)Graph (drawing of the function)Description (describing the relation between x and y)

38Domain of a function

The set of input values for x39Domain of a function

The set of input values for x

http://mathdemos.org/mathdemos/domainrange/domainrange.html40Range of a function The set of output values for y41Range of a function The set of output values for y

http://mathdemos.org/mathdemos/domainrange/domainrange.html42 x is the independent variable and is represented on the horizontal axis (x axis). The x-value is called abscissa.

y or f(x) is the dependent variable and is represented on the vertical axis (y axis). The y-value is called ordinate.

The values of x and y together, written as (x , y) are called the coordinates.

Graph of a function

43Plotting a functionEach axis must be graded with its own scale!

4444Tell pupils that it is most accurate to use a small cross when plotting points on a graph. Stress that when the points do not lie in a straight line we have to decide whether to use a line of best fit, a smooth curve through the points or to join the points together using straight lines. The one we choose depends on the context from which the graph is generated and whether intermediate points have any significance.Plotting a functionNumber of days, dCost in , c123455580105130155003040506070809010011012013014015012345Number of daysCost ()It is most accurate to use a small cross for each point.If appropriate, join the points together using a ruler.Lastly, dont forget to give the graph a title.Cost of car hireEach axis must be graded with its own scale!

4545Tell pupils that it is most accurate to use a small cross when plotting points on a graph. Stress that when the points do not lie in a straight line we have to decide whether to use a line of best fit, a smooth curve through the points or to join the points together using straight lines. The one we choose depends on the context from which the graph is generated and whether intermediate points have any significance.Variations in a functionIncrease A function is increasing if the y-value increases as the x-value increases.

47Increase A function is increasing if the y-value increases as the x-value increases.

48Increase A function is increasing if the y-value increases as the x-value increases.

Ex.: The height of a kid versus his age.49Decrease A function is decreasing if the y-value decreases as the x-value increases.

50Decrease A function is decreasing if the y-value decreases as the x-value increases.

51Decrease A function is decreasing if the y-value decreases as the x-value increases.

Ex.: The height of an elderly person versus his age.

52Increasing and Decreasing intervals Many functions have increasing intervals and decreasing intervals.

53Increasing and Decreasing intervals Many functions have increasing intervals and decreasing intervals.

54Increasing and Decreasing intervals Many functions have increasing intervals and decreasing intervals.

Ex.: The height of a person versus his age.

55Constancy A function is constant if the graph is horizontal, parallel to the X-axis.

xyy = 1All of the points lie on a straight line parallel to the x-axis.56

Example: Graphs parallel to the x-axis

What do these coordinate pairs have in common?

(0, 1), (3, 1), (2, 1), (2, 1), (1, 1) and (3, 1)?The y-coordinate in each pair is equal to 1.Look what happens when these points are plotted on a graph.All of the points lie on a straight line parallel to the x-axis.Name five other points that will lie on this line.This line is called y = 1.xyy = 15757Stress that as long as the y-coordinate is 1 the x-coordinate can be any number: positive negative or decimal.Encourage pupils to be imaginative in their choice of points that lie on this line. For example, (1, 1934792) (1, 56/87) or (1, 0.0000047).

Graphs parallel to the x-axis

All graphs of the form y = c, where c is any number, will be parallel to the x-axis and will cut the y-axis at the point (0, c).xyy = 2y = 5y = 5y = 35858Stress that the graph of y = something will always be parallel to the x-axis. In other words, it will always be horizontal (not vertical like the y-axis). For each graph shown in the example, ask pupils to tell you the coordinate of the point where the line cuts the y-axis.Ask pupils to tell you the equation of the line that coincides with the x-axis (y = 0).

Maxima and minimaMaxima and Minima (local)A function has a relative (or local) maximum at a point if its ordinate is higher than the ordinate of the points around it.

A function has a relative (or local) minimum at a point if its ordinate is less than the ordinate of the points around it.

60Maxima and Minima (local)A function has a relative (or local) maximum at a point if its ordinate is higher than the ordinate of the points around it.

A function has a relative (or local) minimum at a point if its ordinate is less than the ordinate of the points around it.

61Maxima and Minima (local and global)A function has an absolute (or global) maximum at a point if its ordinate is the largest value that the function takes.

A function has an absolute (or global) minimum at a point if its ordinate is the smallest value that the function takes.

62Maxima and Minima (local and global)A function has an absolute (or global) maximum at a point if its ordinate is the largest value that the function takes.

A function has an absolute (or global) minimum at a point if its ordinate is the smallest value that the function takes.

63Trends and periodicityTrendsWe can predict the shape of the graph for some functions for large values of x because they describe events with a very clear trend.

Ex.: Water temperature versus time passed during the heating and turning off the heat.65PeriodicityPeriodic functions are those whose behavior is repeated each time the independent variable covers a certain interval. The length of this interval is called a period.

66Continuity and discontinuityContinuity and discontinuityA function is be continuous when you can plot its graph without lifting your pencil off the paper. A function is be discontinuous when it has discontinuities or jumps on its graph.

.68Continuity and discontinuityA function is be continuous when you can plot its graph without lifting your pencil off the paper. A function is be discontinuous when it has discontinuities or jumps on its graph.

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