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.

1.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 watch the graphs above at desmos.

Exercises

1. Function f add three to every natural number and then squares the result; g squares first every natural number and then, adds three. Write both algebraic expressions. Are they the same? Find the images of 2, 5 and 0 by f and g.

2. If f(x) = 2x - 3, find out the following values:

   1. f(-2) = □

   2. f(□) = 0

   3. f(1) = □

   4. f(□) = 1

3. Given the function y = 3x - 2

   1. Make a table for five different values.

   2. Represent the graph of the function.

4. A driver goes at 100km/h speed. Write a function that relates distance and time spent.

5. Fruits are sold by units in a grocery. An apple costs 0.35 €.

   1. Make a table for the price of 1, 2, 3, 4 and 5 apples.

   2. Represent the graph of the function that turn apples into their prices.

Los siguientes apartados siguen en páginas independientes:

• Features of a function

• Linear functions

• Increasing and decreasing

• Quadratic functions

Exam questions

1. Complete the graph of a function according to some given conditions (domain, continuity, simmetry...).

2. Find f(a) knowing the analityc expression, the graph or a table for f.

3. Real life little problem for a linear function.

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

6. NFDP (New Funny and Different Problem).

Mock exam

1. Completa las siguientes gráficas para que cumplan lo que se dice debajo:

3. A taxi company charges a fixed starting fee of 3.50 euros and 1.20 euros for each kilometre travelled. Write a formula for the total cost C in euros, depending on the distance x in kilometres. Complete a table for x = 0, 2, 5 and 8. How much does an 8 km journey cost? A student has 12 euros. What is the greatest whole number of kilometres they can travel? Explain why this situation can be represented by a straight line.

6. In the school science fair, a very safe potato cannon launches a tiny foam potato. Its height is modelled by: h(x) = -0.5x² + 5x, where x is the horizontal distance in metres and h(x) is the height in metres.

A. Does the parabola open upwards or downwards? What does that mean in this context?

B. Find h(0), h(2), h(5), h(8) and h(10).

C. At what distance does the foam potato reach its maximum height?

D. What is the maximum height?

E. At what distance does it land?

F. Draw an approximate graph. Add a silly but scientific title.

Enlaces

• me DORÉ COCO, la CREMA SICÓ (cancioncilla sobre las características de las funciones). Suno y Francisco Durán.

• 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]

• Historia de las funciones.

• Battlesquare.

• Convierte en gráfica cualquier palabra o expresión.

• Corazones (no son funciones).