UNIT 4: GRAPHS AND FUNCTIONS. - WordPress.com · Unit 1: Graphs and functions. Mathematics 2nd ESO....

Unit 1: Graphs and functions. Mathematics 2 nd ESO. Teacher: Miguel Ángel Hernández.

UNIT 4: GRAPHS AND FUNCTIONS.

Coordinating the plane:

The coordinate grid: In the sixteen century, the French mathematician René Descartes developed a grid of numbers to describe the location of any point in the plane.

In the diagram below we have a horizontal line and a vertical one meeting at the point 0. It is called the coordinate plane ( or grid) or the Cartesian Plane in honour of Descartes:

The horizontal line is called the x-axis.The vertical line is called the y-axis.The point where the two lines meet is the origin.

Plotting points in the plane: Let's plot for example the point (2,1): starting at the origin, we move 2 units to the right along the x-axis; then move vertically up to 1 unit. We describe the location of this point as the ordered pair (2,1). This ordered pair is also called as the coordinates of the point. When giving the coordinates of a point, the x-value comes first. Some points are plotted in the coordinate plane below:

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The four quadrants: The diagram below shows that the x-axis and the y-axis divide the plane into four quarters. Each of this is called a quadrant:

1. Plot these points on the grid shown:A(-5,0) B(-2,-2) C(-4,2) D(0,-1) E(2,-4) F(2,6)

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2. Do the same with these other points:

A 32

,0 B 12

,− 32 C −4,0

D 14

,6 E −52

,−3 F −12

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3. On a coordinate plane, represent the following points and join them in order:

A(3,4) B(3,-1) C(-2,-1) D(-2,4)

What kind of shape do you get?

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4. Do the same with these other points:

A(-2,3) B(2,6) C(6,3) D(6,-2) E(2,-5) F(-2,-2)

What kind of shape do you get?

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Functions:

Definition: A function is a relation between two variables called x and y which:

• x is the independent variable.• y is the dependent variable.• Every x-value is related to one and only one y-value.

Functions can be represented using grids and points. This is important when the function behaviour needs to be visualized. But be careful, because there are graphs which are not functions. For example:

In the second case, for example the x-value -1 has three different correspondences, so, it is not a function.

The set of all real numbers variable x can take such that the expression defining the function is the domain of the function, and the set of all values that the function takes when x takes values in the domain is called the range. For example:

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The graph of a function has to be studied from the left to the right, that is to say, how the y-coordinate varies when the x-coordinates increases.

For example in the last function:Domain= [-4,3]Range= [1,5]

If the graph of the function is going up from left to right, then it is an increasing function.

If the graph is going down from left to right, then it is a decreasing function.

The function of the example is increasing in the intervals [-4,-2] and [-1,2], and is decreasing in [-2,-1] and [2,3].

1. This functions shows the temperature in a city during a day:

Answer these questions:

a) What are the x-variable and the y-variable?

b) What is the domain? And the range?

c) When is the function increasing and decreasing?

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2. The following function shows the number of people in a museum during the hours of a morning:

Answer these questions:

a) What are the x-variable and the y-variable?

b) What is the domain? And the range?

c) When is the function increasing and decreasing?

Another way of seeing the concept of function:

In mathematics we use the word FUNCTION for any rule, where for any input number; there is only one output number. The input and the output numbers may be written as an ordered pair (or a couple).A set of couples will be a function if no two couples have the same first component.

Thus, {(1,4), (2,5), (3,6), (4,7)} is a function but {(2,7), (3,8), (3,9), (4,12)} is not a function as (3,8) and (3,9) have the same first component.

In a set of couples, for example:{(1,1), (2,4), (3,9), (4,16), (5,25)}

The set of first components is the domain:Domain: {1,2,3,4,5}

The set of second components is the range:Range; {1,4,9,16,25}

From this, you can see that a function may be thought of as a machine which processes numbers. In the last example the machine relates every number with its squared.

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Other examples:

Notation for a function:

Notice that all the function that we have seen until now have been studied with a graph which allows us to know some characteristics of the function. But there is a great amount of functions given by an analytical expression or algebraic expression which connects x and y variables algebraically.

We generally use the letter f to represent a function, and also y. If a function tell us to double a number and add 4, 2x+4, the function may be written in any of these ways:f(x)=2x+4ory=2x+4.

When we want to calculate the image of a certain number (the number that corresponds to the given number), we replace x with the number in the algebraic expression of the function.

For example: If we have the function f(x)=2x+4, the image of 3 is f(3)=2·3+4=10

10 is the image of 3: f(3)=10

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INPUT FUNCTION MACHINE

OUTPUT

0 2 3 6

x+2

2 4 5 6

0 1 2 3

2x+3

3 5 7 9

-1 0 2 5

3x

-3 0 6 15


Example:

The function which relates every number with its triple:

– The algebraic expression of the function is: f(x)=3x or y=3x.– To represent the function is convenient to make a table of values:

x y -2 -6 -1 -3 0 0 1 3 2 6

– And finally, we plot these ordered pairs and join them:

1. Write the algebraic expression, make a table of values and draw the graphs of the following functions:

• The function which relates every number with the difference of the number and two.

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• The function which relates every number with the double of the number plus one.

• The function which relates every number with the sum of its double and three.

• The function which relates every number with the opposite of the number minus one.

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2. Plot the graph of the functions whose algebraic expressions are given:

a) y= -x b) y= 12

x

c) y= 1-2x d) y= -x+3

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Continuous and discontinuous functions:

A continuous function is a function whose graph can be drawn without lifting the pen of the notebook (or the chalk from the blackboard). Otherwise, the function is discontinuous. This is only an intuitive definition.

This is the price list of a car park:

Open from 9 to 22 h.First and second hours ….............................Free

Third hour and consecutive or fraction …...1 € each.Daily maximum ….......................................12 €

Drawn the graph which relates park timing to its price. Is it a continuous function?

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Axes intercepts:

Given the graph of a function:

• An x-intercept is a point where the graph of the function meets the x-axis. x-intercepts are found letting y be 0 in the algebraic expression of the function.

• An y-intercept is the point where the graph of the function meets the y-axis.y-intercepts are found letting x be 0 in the algebraic expression of the function.

Example: Function y=2x+4

• The graph of this function intercepts the x-axis on the point (-2,0) because:

y=0⇒2x4=0⇒2x=−4⇒ x=−42=−2⇒ x=−2

• The graph of this function intercepts the y-axis on the point (0,4) because:

x=0⇒ y=2⋅04=4⇒ y=4

Find the x and y-intercepts of the following functions:

y=x+2 y=3x+3 y=2x-6 Represent their graphs and write these points.

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Increasing and decreasing functions. Maximum and Minimum:

• If the graph of a function is going up from left to right, it is an increasing function.• If the graph is going down from left to right, it is a decreasing function.• If a function changes from increasing to decreasing when it passes through a point, this

point is called a maximum of the function.• If a function changes from decreasing to increasing when it passes through a point, this

point is called a minimum of the function.

Example:

The function is increasing in the intervals: [0,2] and [4,6].The function is decreasing in the intervals: [2,4] and [6,8].There are two maximum points: (2,6) and (6,4) and there is one minimum point (4,1).

1. Study the intervals where the following graph is increasing or decreasing, and write the maximum and minimum points:

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2. The following graph shows the present people (in thousands) in a shopping centre during a day.

Determine the intervals of increasing and decreasing, and the maximum and minimum points.

3. The following graph shows the number of people who have lived in a city during some years.

Determine the intervals of increasing and decreasing, and the maximum and minimum points.

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Function of direct proportionality:

A function of direct proportionality or a linear function is a relation between two direct proportionality magnitudes.

• Its algebraic expression is y=mx , where m is the constant of direct proportionality.• Its graph is a straight line that passes trough the origin.• The constant of proportionality is called the gradient or slope of the straight line.

➔ If m is positive, the function is increasing.➔ If m is negative, the function is decreasing.

1. Represent the following functions of direct proportionality, write down their gradients and study if the functions are increasing or decreasing.

y=2x y=3x y=x

y=-2x y=-3x y=-x

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2. If a can of beer costs 0,45 € at a supermarket:a) Make a table of values to relate the price and the number of cans you buy.b) Write the algebraic expression of the function.c) Represent the graph of the function.

Function of inverse proportionality:

A function of inverse proportionality is a relation between two direct proportionality magnitudes.

• Its algebraic expression is y= kx , where k is the constant of inverse proportionality.

• Its graph is a curve called hyperbola.

Examples:

The graph of the functions of inverse proportionality:

y=1x

y=−2x

x y x y

1 1 1 -2 2 1/2 2 -1 3 1/3 3 -2/3 -1 -1 -1 2 -2 -1/2 -2 1 -3 -1/3 -3 2/3

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1. If a car goes from the city A to the city B with a constant speed, and the distance between the cities is 240 km:

a) Complete the table below relating speed and time:Speed (km/h) 20 40 60 80 100 120Time (h)

b) Write the algebraic expression of the function speed-time.

c) Represent the function.

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2. If a car goes with a constant speed of 100 km/h:

a) Complete the table below relating time and distance:Time (h) 1 2 3 4 5Distance (km)

b) Write the algebraic expression of the function speed-time.

c) Represent the function.

3. Determine the algebraic expression of a function of direct proportionality that passes through the point (2,4). Represent the function.

4. Determine the algebraic expression of a function of direct proportionality that passes through the point (-1,3). Represent the function.

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Keywords:

graph=gráfico, gráfica.function=funciónCoordinate grid/Coordinate plane=Plano de coordenadasCartesian plane=Plano Cartesianox-axis=eje X, eje de abscisasy-axis=eje Y, eje de ordenadasorigin=origen de coordenadaspoint=puntocoordinates=coordenadasordered pair=par ordenadoquadrant=cuadranteindependent variable=variable independientedependent variable=variable dependientedomain=dominiorange=recorrido, rango, conjunto imagenIncreasing function=Función crecienteDecreasing function=Función decrecienteAnalytical expression/Algebraic expression=Expresión analítica, expresión algebraica.image=imagentable of values=tabla de valorescontinuous function=función continuadiscontinuous function=función discontinuaaxis intercepts= puntos de intersección o puntos de corte con los ejesx-intercept=punto de intersección (o de corte) con el eje xy-intercept=punto de intersección ( o de corte) con el eje ymaximum point=punto máximo /maximum=máximo /maxima=máximosminimum point=punto mínimo /minimum=mínimo /mimima=mínimosfunction of direct proportionality=función de proporcionalidad directaconstant of direct proportionality=constante de proporcionalidad directagradient/slope=pendientestraight line=rectafunction of inverse proportionality=función de proporcionalidad inversaconstant of inverse proportionality=constante de proporcianalidad inversahyperbola=hipérbola.

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UNIT 4: GRAPHS AND FUNCTIONS. - WordPress.com · Unit 1: Graphs and functions. Mathematics 2nd ESO....

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