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ACTIVITY 8 vertex of a parabola

Activity description identifies the student and the vertex of a parabola graphically. Now find the vertex with two different methods completing the trinomial perfect square and by formula. Using GeoGebra graph the function and point to the vertex and the methods studied in class will find the vertex by the method indicated. The following activities. Development of the activity

The vertex of a parabola is represented by V (h, k), and can be obtained algebraically and graphically.
Algebraically
know that quadratic functions are written in the general form y = ax ² + bx + c can be easily obtained by the method of completing the square trinomial perfect place to express it in standard form y = a (xh) ² + k . Using Graphically

GeoGebra can graph a quadratic function and identify the vertex.
determines the vertex function f (x) = 2x ² - 8x + 5 by the method of completing the square and then check your results with the help of GeoGebra noting the location of the vertex.
If we know that h = (- b/2a) and k = c - (b ² / 4a) can apply these formulas to get the vertex.
determines the vertex of f (x) = x ² - 2x + 3, if we know that c =______. =_____, b =______ Then we apply the formulas:
V (- b/2a, c - (b ² / 4a)
V (- _____/2_______, _______ - (_____²/ 4_______))
V (-______, 3-______) V
(______, 2)
Determines the top of the parables by the method of completing the square and check your result in GeoGebra.
a) f (x) = x ² - 10x - 3
b) g (x) = 4x ² - 12 x + 5
c) h (x) = 2x ² - 6 x + 3
determines the vertex of the parabolas using the second method and check the result in GeoGebra pointing your result.
a) f (x) = 3x ² - 2x + 1
b) g (x) = 27x ² + 12 x - 7
c) h (x) = x ² - 16 x - 63

ACTIVITY 9
Description of Activity
The student will practice everything learned during the teaching sequence solving the following activities: Development

activity determines the characteristics of each of the following quadratic functions, finding solutions, the behavior of the graphs comparing with f (x) = x ² with the help of GeoGebra. For each of the equations fill out a questionnaire as follows: Ratio
cuadrático_______________
Towards the end where they open the concavity es_______________________________
ramas_____________________
It has a maximum or mínimo_________________
The maximum or minimum value is______________________
axis equation symmetry is x = h, then the axis of symmetry is x =__________
The value that corresponds to the maximum or minimum does not supply the value of k.
This is f (h) = k so it _________. Vertex is V
(____,_____)
a) f (x) = 2x ² + 8x - 5
b) g (x) = 1/4x ² + 8 x + 3
c) h (x) = x ² + 4 x
d) f (x) = 2x ² - 8x
e) g (x) =-x ² - 6 x +8
f) h (x) = 2x ² - 12 x + 19
g) f (x) = ¾ x ² - 6x + 2
h) g (x) =-3x ² - 18 x + 26
i) h (x) = 4x ² - 8 x - 7

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1 Latest Activity Activity 6 and 7 (SEQUENCE LEARNING UNIT 1 MATE 2)

ACTIVITY 6 and 7
ANALYSIS OF PERFORMANCE PARAMETERS OF THE FUNCTION QUADRATIC

Description of Activity Students plotted in GeoGebra activity exercises 6 and record your observations. After watching the behavior of the graphs from previous years check the behavior of quadratic functions of activity 7: Develop
Exercises
activity activity 6
In a Cartesian coordinate system itself performs the following functions graphs and scores your observations by comparing them with x ².
f (x) = x ², f (x) = ¾ x ² f (x) = 4 x ².
Comments: ________________________________________________________________________________________________________________________________________________
Now do the same but with f (x) = x ² + 5, f (x) = x ² - 5 and compared with the graph of f (x) = x ².
Comments: ___________________________________________________________________________________________________________
Make the same but now f (x) = (x + 13) ² f (x) = (x - 13) ² comparing again with f (x) = x ².
Comments: ________________________________________________________________________________________________________________________________________________
Exercises
activity 7 analyzes the procedure involves the following functions. For analysis of the behavior of the quadratic function on f (x) = x ².

a) f (x) = - (x - 2) ²
b) k (x) = ¼ (x + 5) ² - 3
c) h (x) = 3 (x +7) ² +24
d) g (x) = - ½ (x - 13) ² -11
e) i (x) = (x + ¾) ² + 6

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TASK 2 MATE 1 UNIT 1 UNIT 1

Clear and write the following functions depending on x. Now, graph and identify the solutions of the equations:
a) 1/6x ² + 1 / x = 1
b) 4 - x = x +2 / x - 2
c) 7 x ² = 8x - 2
Determine graphically the solutions the following quadratic functions:
a) f (x) = 2x ² + 8x -5
b) g (x) = X ² + 4
c) h (x) = - x ² - 6x
d) k (x) = 3x ² + 1
e) f (x) = 3x ² + 9x + 1
f) f (x) = (1 / 4) x ² + 8x + 3
g) g (x) = 2x ² - 8x
h) h (x) = - 5x ² - 2
i) f (x) = x ² - 2x - 3
j) h (x ) = (1 / 2) x ² - 6x

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SEQUENCE LEARNING MATE 2

MATH TEACHING SEQUENCE 1
Description of Activity:
was investigated in printed and electronic concepts of activity 1. His research verify how reliable are websites compared with those found in print. So they must take both the Internet literature as the book where you found it. ACTIVITY 1

Research on the Internet and compare it with some books concepts as
Function Function Types


quadratic function of a quadratic function Grafica
components of the graph of a quadratic function. ACTIVITY 2


Activity description examples will be proposed that involve different types of functions, in which students have to say whether or not involving a quadratic function. If it is to propose a way to solve it. At least they will have to take a stake in each proposal.
Problems
development activity leading to a quadratic function:
From a pedestrian bridge, with a height of 5.50 m, vertically dropped ball. How long will it take the ball hit the ground?
It uses a formula that relates the distance, time and gravity in this case is also a very important factor: d = ½ g t ² if we know that g = 9.81 m / sec ² and time t unknown.
A complete what is needed to reach a quadratic function. ________=
½__________ t ² if t = x
Transposing terms, the equation becomes:
_______x ²-_______= 0
The volume of a cylinder is 1000〗 〖cm ^ 3. Express the total area of \u200b\u200bthe cylinder as a function of the radio and make your graphic
extra Documents:
Video: "Application of the quadratic function" http://www.youtube.com/watch?v=TF2_IjxOtyY accessed August 13, 2010 . Activity 2
exercise involving quadratic functions. ACTIVITY 3


Activity description
The student conduct GeoGebra using graphs of a quadratic function and compare with other quadratic function, observe and perform their activity exercises 3. Then graph a quadratic function and a linear function, doing the exercises activity 4.

Development of activity
FINANCIAL ACTIVITY 3
GRAFICA of a quadratic function
in GeoGebra enter the data into the entrance area and draw the graph of y = x ², and answer the following questions:
In this case if we know the equation a quadratic function is given by y = ax ² + bx + c. What is the value of a? ________
The apex is at the point (_____,_____)
The socket is ________________ ________________
open branches
The axis of symmetry is x = __________
has a maximum or minimum value _______________
Now construct the graph of y = - x ², and answer: What is the
value of a? ___________
The apex is at the point (_____,_____)
The socket is ________________ ________________
open branches
The axis of symmetry is x = __________
has a maximum or minimum value
_______________ Make comments you have regarding the two graphs.
_________________________________________________________________________________________________________________________________________ determines the elements of the following quadratic functions with the help of GeoGebra:
f (x) = 2x ² + 8x -5
g (x) = x ² +4
h (x) =- x ² - 6x
k (x) = 3x ² + 1
f (x) = 3x ² + 9x + 1
FINANCIAL ACTIVITY 4
quadratic function FUNCTION LINEAR
in GeoGebra follow these graphics in a single Cartesian coordinate system:
1) y = x ² y = x (in input introduces first y = x ² the square you can put the bar after login, then y = x) using the spreadsheet performs the tables and graphs made separately to obtain the behavior of each table.
2) y = x ² +1 and y = x + 1 does the same as in the preceding paragraph.
now makes the observations on the behavior of each graph and the difference in each of the tables got.
Remarks Table 1 y = x ² y = x
______________________________________________________________________________________________________________________________________________
Observations Table 2 y = x ² + 1 y = x + 1
______________________________________________________________________________________________________________________________________________
ACTIVITY 5
Roots of quadratic equation in Association with the quadratic function
The graph of a quadratic function allows visually find the solution of the quadratic equation.
Using the worksheet get the table of values \u200b\u200band graph of the following quadratic functions:
f (x) = x ² + 2x -8
g (x) = x ² - 3x -10
Note that each graph intercepts the x axis at two points indicates the value of each point:
x_1 =__________ x_2 = __________ and __________ and
x_1 = x_2 =__________
These values \u200b\u200bare the roots or solutions of each of the previous quadratic functions. Using GeoGebra
determines the roots of the following quadratic functions and draw the roots in the graph.
f (x) = x ² + 3x -10 = ______ and x_2 x_1 =_______
g (x) = x ² + 2x +1 x_1 = x_2 = ______ and ______
h (x) = x ² - x - 30 x_1 = ______ and x_2 = ______
k (x) = x ² + 2x + 3 x_1 = ______ and x_2 = ______
f (x) = x ² + 2x - 3 x_1 = ______ and x_2 = ______
h (x ) = 0.5x ² - 6x x_1 = x_2 = ______ and ______