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Introduction
Introduction
“What happens near these vertical lines?”
“Does the graph go up or down here?”
“Where are the hills and valleys?”
“Is the graph continuous?”
“Are there parts that seem symmetrical?”
“Does it bend upwards or downwards?”
“Does it approach a line?”
“Can you find regions where it grows?”
“Which part looks dangerous for a skateboarder?”
“Where would water accumulate?”
“If a person walked from left to right, when would they climb or descend?”
Domain and Range
Domain and Range
Domain
Domain
The domain is the set of all possible values of x.
In other words: “Which x-values can we use?”
Example:
A function describing the age of a person could not have negative ages.
So the domain would start at 0.
On a graph, to find the domain: look from left to right
On a formula:
If the function has a fraction, we must avoid values that make the denominator zero.
If the function has a square root, we must avoid negative numbers (in real numbers). f(x) = √x, the domain is all numbers that are greater than or equal to 0, because we can't take the square root of a negative number (in real numbers). So, the domain is: x ≥ 0 or [0,∞).
Range
Range
The range is the set of all possible values of y.
In other words: “Which y-values appear in the graph?”
On a graph, to find the range: look from bottom to top.
Exercise 1 – Real-Life Situations
Write the domain of each function.
The height of a baby as a function of age.
The amount of petrol in a car tank as a function of kilometres travelled.
The score of a video game as a function of playing time.
The area of a square as a function of the side length.
The temperature of a pizza as a function of time after taking it out of the oven.
Exercise 2 - looking at the formula
Find the domain of the functions to the right (click to see their graphs)
Exercise 3 – Explain the restriction
For each function:
Find the domain.
Explain WHY some values are not allowed.
Find the domain of the functions to the right (click to see their graphs)
Axis interceptions
Axis interceptions
The intercepts of a graph are the points where the graph crosses the coordinate axes.
Y-intercept
Y-intercept
The y-intercept is the point where the graph crosses the y-axis.
To find it, we use:
x=0x=0x=0
Example:
y=2x+3y=2x+3y=2x+3
If x=0x=0x=0:
y=2⋅0+3=3y=2\cdot 0+3=3y=2⋅0+3=3
So the y-intercept is:
(0,3)(0,3)(0,3)
X-intercepts
X-intercepts
The x-intercepts are the points where the graph crosses the x-axis.
To find them, we use:
y=0y=0y=0
Example:
y=x−4y=x-4y=x−4
If y=0y=0y=0:
0=x−40=x-40=x−4 x=4x=4x=4
So the x-intercept is:
(4,0)(4,0)(4,0)
Important idea
Important idea
On the y-axis, xxx is always 0.
On the x-axis, yyy is always 0.
So, to study the intercepts:
Y-intercept→x=0\text{Y-intercept} \rightarrow x=0Y-intercept→x=0 X-intercepts→y=0\text{X-intercepts} \rightarrow y=0X-intercepts→y=0
Intercepts are useful because they help us draw the graph and understand where the function starts, crosses, or touches the axes.
Simmetry
Simmetry
EVEN (PAR): f(-x) = f(x), about the Y-axis;
ODD (IMPAR): f(-x) = -f(x), about the origin.
Periodicity
Periodicity
(skip for now ;-)).
Continuity/Continuidad
Continuity/Continuidad
Look at the following two graphs:
What differences do you notice between them?
Perhaps the most obvious one is that the first graph has two separate parts, while the second one has only one.
Well, continuity is a key concept in functions and has a simple explanation:
if the graph can be drawn in one single stroke, the function is continuous.
Say which of the graphs seen so far in this lesson are continuous.
Derivability
Derivability
For now, just think of it as the "softness" of its graph. Derivability ⇒ Continuity
Growth
Growth
The following table shows the evolution of the population in Granada (city) from the beginning of the 20th century until the last census in 2011 (population censuses are carried out every ten years).
(Source: National Statistics Institute (INE))
Represent this table graphically using an appropriate scale (x-axis: years; y-axis: population).
In which year was the population the highest? And the lowest?
If we only consider the period between 1940 and 1991, in which year was the population the highest? And the lowest? Do these answers match those from the previous question?
Write in your notebook the periods in which the population increased and the periods in which it decreased.
Of all the growth periods, which one was the fastest? Justify your answer. Do the same with the periods of population decrease.
In the year 1940, was Granada's population increasing or decreasing? And in 1960?
Imagine a person running along the following black line:
There are moments when the person goes up and others when they go down. That blue man goes up and down from left to right.
Well, if that line were the graph of a function, we would say that the function is increasing in the parts where the person goes up — that is, the orange sections — and decreasing in the parts where the person goes down — the green sections.
We refer to parts of a function by the section of the x-axis they occupy (just pay attention to the x-values). Therefore, in the previous example — reproduced below with the scale included — we would say that the graph is decreasing from the beginning of the drawing to −4; increasing from −4 to 2; decreasing again from 2 to 6; and, from that point onwards, increasing indefinitely.
Maximums and Minimums
Maximums and Minimums
There are points on the graph where the function changes from increasing to decreasing, or vice versa. These are the hills and valleys that can be seen in the following picture.
The point where the little tree is, (2, 2), is higher than all the nearby points around it. There is a maximum of the function there.
The points where there is water, (-4,-8) and (6,-2), are lower than all the nearby points around them. They are called minimums.
There are places on the graph that are higher than the little tree (for example, where the branch is). That is why the point where the tree is called a relative maximum.
For similar reasons, the point (6,-2), where the light blue water is, is called a relative minimum.
However, there is no point on the graph lower than (-4,-8). There we say that the function has an absolute minimum.
In the same way, an absolute maximum is a point on the graph that is higher than all the others.
In the example drawing there is no absolute maximum because we assume that the graph continues upwards indefinitely.
For now, we will find the maximums and minimums of a function by looking at its graph, although there are analytical methods to do this as well.
Exercises
Exercises
Observe the following graphs and study their increasing or decreasing behavior, and their maximums or minimums. Also indicate whether they are relative or absolute.
Curvature: Convexity, Concavity and Inflection Points
Curvature: Convexity, Concavity and Inflection Points
Graphs do not only go up or down. They can also bend in different ways.
Concave Functions
Concave Functions
A function is concave when the graph bends like a smile 🙂. The curve opens upwards.
Example: y = x²
A convex graph usually has a shape like this: ∪
Convex Functions
Convex Functions
A function is convex when the graph bends like a frown ☹️. The curve opens downwards.
Example: y=−x²
A concave graph usually has a shape like this: ∩
Inflection Points
Inflection Points
An inflection point is a point where the graph changes its curvature.
That means the graph changes from convex to concave, or from concave to convex.
Example: y=x³
This graph changes curvature at (0,0). So (0,0) is an inflection point.
Important Note
Important Note
A function can be:
increasing and convex,
increasing and concave,
decreasing and convex,
decreasing and concave.
So growth and curvature are different ideas.
For example, a graph can be going up but bending down, like a ball that is still rising but slowing down.
Asymptotes and Infinite Branches
Asymptotes and Infinite Branches
Sometimes, a graph gets closer and closer to a line but never actually touches it. These special lines are called asymptotes.
Vertical Asymptotes
Vertical Asymptotes
A vertical asymptote is a vertical line that the graph approaches infinitely.
Near that line, the function goes very high or very low very quickly.
Example:
the graph may go upwards forever,
or downwards forever,
when it gets close to a certain x-value.
The function is usually not defined on that line.
Horizontal Asymptotes
Horizontal Asymptotes
A horizontal asymptote is a horizontal line that the graph approaches as we move far to the left or far to the right.
The graph gets closer and closer to that height, even if it never reaches it.
Infinite Branches
Infinite Branches
An infinite branch is a part of the graph that continues forever.
For example:
upwards,
downwards,
to the left,
or to the right.
Graphs with asymptotes often have infinite branches.
Inflection points. Where the curve changes its curvature; U to ∩ or ∩ to U.
Asymptotes. An asymptote is a line that a curve approaches but never touches or crosses.
Ejercicios sobre gráficas
Ejercicios sobre gráficas
The following graph shows the height of the water in a tank after a week.
Which day did the water reach the maximum height? What height was that?
When did it reach the minimum height? What height was that?
Say the intervals when the level of the water decreased.
What does the point where the line touches the horizontal axis mean?
3. Explain if the following function is continuous. Point at the intervals where the function is increasing, decreasing or constant.
4. Describe the following graphs using this vocabulary:
The function is continuous or discontinuous.
The function is increasing in the interval ...
The function is decreasing in the interval ...
The function is constant in the interval ...
The function has a minimum at the point ...
The function has a maximum at the point ...
The intersection of the curve with the x-axis is ...
The intersection of the curve with the y-axis is ...
5. Da las coordenadas de un máximo relativo, un mínimo relativo, un máximo absoluto, un mínimo absoluto, indicando a qué gráfica corresponden.
7. Indica los intervalos de concavidad y convexidad de cada una de las gráficas.
8. Calculate f(0) and f(1) for each of the functions represented by those graphs.
9. Which of the last graphs has two máxima?
8. Do the test at the bottom of this webpage about evaluating functions.
9. Preguntas sobre las siguientes gráficas.
10. Describe the following function.
Función de proporcionalidad directa
Función de proporcionalidad directa
El coordinador de un viaje de intercambio al Reino Unido desea ofrecer a sus alumnos una tabla de conversión de euros a libras. Elabora esa tabla con cinco valores y sus conversiones, teniendo en cuenta que 1 euro equivale, aproximadamente, a 0.75 libras. ¿Crees que la relación euros - libras es una función? Representa los valores en un sistema de ejes cartesianos y conecta los puntos. ¿Qué forma tiene la gráfica? ¿Puedes encontrar su fórmula?
You can buy apples by 2 euros a kilogram. Fill the following table:
3. Observa el muelle de esta web. Pon el manejador azul a 100 (mide la resistencia del muelle). Elabora una tabla con algunos valores diversos para la fuerza y la longitud correspondiente del muelle. A partir de esos valores, elabora una fórmula que dé la fuerza aplicada en función de la longitud del muelle.
Pounds and euros, weight and price, force and length... are directly proportional magnitudes. You get pounds by multiplying the amount of euros by 0.75; you get prices if you multiply the number of kilograms by 2; you get the increase of the length of the spring when you multiply the force by 100. So 0.75, 2 or 100 are in this situations constants of proportionality. Those examples can be represented by the following formulas L = 0.75·E, P=2·W, L=100·F.
Todas ellas tienen una expresión similar y = m·x. Es la llamada función de proporcionalidad directa.
4. Represent y = 0.5x, y = x, y = 2x and y = 3x. What are the similarities and the differences among all those graphs? You can use geogebra to draw better graphs.
5. Represent y = -0.5x, y = -x, y = -2x and y = -3x. What happen now?
6. Use this site to play with m, in a linear function. Type y=mx inside the box. Find out the meaning of m in that equation.
7. Susan has bought 10 tickets to the cinema by 75 euros. Make a table that represents the number of tickets and their price. Write the corresponding formula.
8. Represent the following functions: y = x, y = -x, y = 2x, y = -2x.
9. The constant of proportionality between two magnitudes is -3. Write the formula of the function. Draw it in a cartesian system.
10. Who's who: play with lines. It's based on the famous "Who's who" board game. In the first rounds, the teacher thinks of one of the lines shown on the whiteboard and the students will have to guess what you are thinking based in your Yes-No answers to their questions. They can decide their questions in pairs for these first rounds. Let them everybody play at least once. And, of course, if someone ask a question in his/her turn, he or she has a try to guess the line. They are allowed to ask anything, but it would be easier if they ask about the slope of the line and somethething more (a point, the y-intercept...). When they get the hang of it they can play one against the other. The slopes are: a: -1, b: -1/2 or -0.5, c: 1/2 or 0.5, d: 0, e: -4, f: -2, g: 2, h: 1, i: 1, j: -1/2, k: -1, l: 1/5 or 0.2, m: 1/5, n: -1, o: 0, p: 1
11. Find out the weight of the objects red, green and yellow from this web site.
Linear functions
Linear functions
Mira el tiempo en Minneapolis. Usa la configuración del sitio para averiguar la relación entre grados Farenheit y Celsius. Para ello, construye una tabla con varios valores y dibuja la gráfica correspondiente. Puedes usar la calculadora de Desmos para ello. Ten en cuenta que, a veces, se redondea el resultado.
F = 1.8C + 32
At which Farenheit temperature does water freeze into ice? What is the boiling point of water in Farenheit scale?
Represent y = 2x -1, y = 2x, y = 2x + 1. What are the similarities and the differences among all those graphs?
La tarifa de una empresa de telefonía establece lo siguiente: 5 céntimos por minuto en llamadas a cualquier operador, más 20 céntimos por establecimiento de llamada. Escribe una fórmula, una tabla con algunos valores y dibuja la gráfica correspondiente para la función que te da el precio de la llamada en función de los minutos hablados.
A ciclyst goes during 280 km at a constant speed of 40 km/h.
Make a table of the journey with five reference points.
Write the algebraic expression of the function.
Draw its graph and describe it.
To convert centimeters into inches you have to multiply by 2 and divide the result by 5. If x represents cm and y, in
Write y as a function of x.
Make a table with some values.
Draw the graph of the function.
Estimate the value of 3 cm according to the graph.
y = mx + n
m = pendiente / slope (mide lo inclinada que está la recta).
n = ordenada en el origen / y-intercept (determina el punto por el que la recta pasa cuando toca al eje de la y, esto es, si la recta está más arriba o más abajo).
Función de proporcionalidad inversa
Función de proporcionalidad inversa
No hay tiempo de verla. No entra en el examen.
Review
Review
Represent in a cartesian coordinate system the following:
A point A whose abscissa is 2 and ordinate -1.
A point B whose abscissa is 4 and ordinate is -3.
The function f convert every natural number by adding 4 units to it and raising the result to the square. The function g associates every natural number with its square and adds 4 units to the result.
Write their equations.
Are f and g the same functions?
Calculate the images of 2, 5 and 0 by either f and g.
Fill the blanks knowing that f(x) = 3x - 2:
f(-2) = ░
f( ░ ) = 0
f(0)= ░
f( ░ ) = 1
Given the following function: y = 3x - 2
Make a table with five values and its images.
Represent the graph of the function.
A car has a speed of 100 km/h. Write a function that makes the relationship between the distance and the time.
Invéntate una función definida con palabras. Debe transformar números en otros números y su dominio debe ser, al menos, el conjunto de los números naturales. Intenta que no use demasiadas operaciones matemáticas. Un buen ejemplo es la que vimos al principio del tema que transformaba cada número en el número de letras que tenía al escribirlo correctamente en inglés. El ganador o ganadores del concurso será la función que sea más difícil de adivinar al mostrar cinco resultados en una tabla y esté expresada en términos más sencillos.
Explica las variables de estos gráficos. Construye tus propios gráficos del mismo tipo.
Un comerciante tiene que vender 1000 kg de naranjas al final de la temporada. Cada día que pasa, pierde 40 kg (se le pudren), aunque el valor del kilo se incrementa 15 céntimos. Describe la situación y decide por él cuándo tiene que vender. (Fuente).
REPASA LOS CONTENIDOS DEL TEMA CON EL SIGUIENTE KAHOOT!
Bonus content
Bonus content
Links
Links
Las funciones y la hoja de cálculo.
Learn the relationship between direct proportion and functions with this page.
Notes:
[1] Many of these games are adapted from the ones in the course “Interactive Teaching - Using Educational Games in order to Enhance Learners’ motivation”, by Felicia Dimulescu, from Eruditus (Erasmus + partner).
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