U10. Functions

Temporalización: 14 horas

• 1. Functions and correspondences / Correspondencias y funciones

  2. Features of functions / continuity, simmetry, periodicity, increasing and decreasing intervals, maxima & minima, concavity-convexity.

  3. Funciones lineales

  4. Ecuaciones de la recta y posiciones relativas de las rectas

  5. Funciones cuadráticas

  6. Funciones en la vida real

Situación introductoria

Para entrar en una casa de Fortnite (con un "loot" impresionante) es necesario pulsar un número. Ese número depende de otro número que se muestra en pantalla. Agazapado tras un arbusto, he visto que cuando la pantalla mostraba 8809, la persona que entró tuvo que teclear 6. Otras informaciones que pude desvelar espiando a más gente fueron: 7111 # 0, 2172 # 0, 6666 # 4, 1111 # 0, 3213 # 0, 7662 # 2, 9313 # 1, 0000 # 4. Es mi turno y en pantalla aparece 2581. ¿Qué número debo teclear para acceder a la casa? 

Since the values of the first set (D) and their "transformed" values can vary, they are called "variables." The elements of the initial set form the independent variable; those of the final set, the dependent variable or image.

The initial set is called the Domain, and the final set is called the Range.

We represent the function like this: f(x) = y, where x is the independent variable and y is the dependent variable. In the previous diagram, 2 is transformed into 4. If the function is called f, we say f(2) = 4.

• Example 1 (words): The function h, which transforms each number into the number of holes it has when written in the Arial font. What does 12 get transformed into? What is the value of h(4600)? Build a table with the image of ten different numbers.

• Example 2 (table): The function p that assigns to each country its Mathematics score from the 2018 PISA report. Calculate p(Spain).

• Example 3 (formula): The function a, defined by the formula y = x² + x + 1. Calculate a(-2) and a(3). Is there any number that gets transformed into 0?

• Example 4 (graph): A line in a cartesian coordinate system that goes from left to right.

Look at the drawing above and calculate f(0), g(1) and q(–2) (click to enlarge).

Also calculate p(0), q(–2), s(–3), b(–1), p(5), f(–1), q(3), s(0), b(0), q(–1), f(1), s(4), b(1) and f(2).

(*) Click here to download the GeoGebra file that contains all the graphs above.

Exam questions

1. Describe some features of functions in a graph: continuity, simmetry, increasing or decreasing, maximum, minimum, concavity, convexity, inflection points, or complete a drawing to achieve something of these features.

2. Domain of a function. Find f(a), knowing a and the analityc expression of f.

3. Problemilla de funciones lineales.

4. Describe some features of functions in a graph: continuity, simmetry, increasing or decreasing, maximum, minimum, concavity, convexity, inflection points, or complete a drawing to achieve something.

5. Interpretar la pendiente de una recta. Posiciones relativas y puntos de corte de dos rectas.

6. Match graphs and analityc expressions (lines and parabolas).

7. PNDD.

Enlaces

• Function calculator. Hace gráficas, evalúa, calcula límites, derivadas, integrales...

• Functions (Desmos).

• ¿Cualquier tiempo pasado fue mejor?

• Cabezas juntas numeradas

• Graphing stories.

• Definition of function (Math is fun).

• Guess the function game. The graphs of several functions are mixed in a bag.

• The students are divided into groups and one by one (in turns) have to come pick up a function and describe it to the group without saying the name or the equation. Time: 1 minute per person. If the group guess they have one point and can pick up another one, if they don’t guess, then the group has minus one point and wait for the next turn. [1]