Matematika 11 Ushtrime Te Zgjidhura Pegi 132 |VERIFIED|
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Matematika 11 Ushtrime Te Zgjidhura Pegi 132: A Complete Guide
If you are looking for a comprehensive and easy-to-follow guide to solve the math problems from page 132 of Pegi 11, you have come to the right place. In this article, we will show you how to tackle each problem step by step, with clear explanations and examples. Whether you are a student or a teacher, this guide will help you master the concepts and skills covered in this page.
Pegi 11 is a math textbook for high school students in Albania, written by Steve Fearnley, June Haighton, Steve Lomax, Peter Mullarkey, James Nicholson and Matt Nixon. It covers topics such as algebra, geometry, trigonometry, statistics and calculus. The book is divided into two parts: Part I and Part II. Page 132 belongs to Part II, which focuses on advanced algebra and functions.
What are the problems on page 132?
Page 132 contains 14 problems that test your understanding of quadratic functions and equations. A quadratic function is a function of the form f(x) = ax + bx + c, where a, b and c are constants and a ≠ 0. A quadratic equation is an equation of the form ax + bx + c = 0, where a, b and c are constants and a ≠ 0.
The problems on page 132 ask you to do the following tasks:
Find the roots of a quadratic equation by using the quadratic formula or by completing the square.
Find the vertex, axis of symmetry, y-intercept and x-intercepts of a quadratic function by using its standard form or vertex form.
Sketch the graph of a quadratic function by using its features and transformations.
Solve word problems involving quadratic functions and equations.
How to solve the problems on page 132?
To solve the problems on page 132, you need to apply the following formulas and methods:
The quadratic formula: If ax + bx + c = 0, then x = (-b ± √(b - 4ac))/(2a).
Completing the square: To rewrite a quadratic expression of the form x + bx + c as a perfect square plus a constant, you need to add and subtract (b/2). For example, x + 6x + 5 = (x + 6x + 9) - 4 = (x + 3) - 4.
The standard form of a quadratic function: f(x) = ax + bx + c, where a, b and c are constants and a ≠ 0. The standard form tells you the y-intercept of the function (c) and whether it opens up (a > 0) or down (a < 0). To find the x-intercepts, you need to solve f(x) = 0.
The vertex form of a quadratic function: f(x) = a(x - h) + k, where a, h and k are constants and a ≠ 0. The vertex form tells you the vertex of the function ((h,k)) and whether it opens up (a > 0) or down (a < 0). To find the x-intercepts, you need to solve f(x) = 0.
The axis of symmetry of a quadratic function: The vertical line that passes through the vertex of the function. Its equation is x = h, where (h,k) is the vertex.
The transformations of a quadratic function: If you compare two quadratic functions of the form f(x) = a(x - h) + k, you can see how they are related by transformations. For example, if you compare f(x) = (x - 2)
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How to use this guide?
This guide is designed to help you solve the problems on page 132 of Pegi 11 in a systematic and efficient way. For each problem, we will provide you with the following information:
The problem statement and the answer.
The steps to solve the problem, with detailed explanations and examples.
The key concepts and formulas that are used in the problem.
The tips and tricks to avoid common mistakes and errors.
You can use this guide as a reference, a study tool, or a practice resource. You can follow along with the steps, check your answers, review the concepts, or try to solve the problems on your own before looking at the solutions. You can also use this guide to prepare for your exams or tests, as it covers the most important topics and skills that you need to know.
Problem 1
Problem statement: Solve the equation x - 5x - 14 = 0.
Answer: x = -2 or x = 7.
Steps:
Use the quadratic formula to find the roots of the equation. Recall that if ax + bx + c = 0, then x = (-b ± √(b - 4ac))/(2a).
In this case, a = 1, b = -5, and c = -14. Plug these values into the formula and simplify.
x = (-(-5) ± √((-5) - 4(1)(-14)))/(2(1))
x = (5 ± √(25 + 56))/2
x = (5 ± √81)/2
x = (5 ± 9)/2
x = (5 + 9)/2 or x = (5 - 9)/2
x = 14/2 or x = -4/2
x = 7 or x = -2
Key concepts and formulas:
A quadratic equation is an equation of the form ax + bx + c = 0, where a, b and c are constants and a ≠ 0.
The quadratic formula is a method to find the roots of a quadratic equation. It states that if ax + bx + c = 0, then x = (-b ± √(b - 4ac))/(2a).
The roots of a quadratic equation are the values of x that make the equation true. They are also called the solutions or the zeros of the equation.
To use the quadratic formula, you need to identify the values of a, b, and c from the equation, and plug them into the formula. Then, you need to simplify the expression under the square root (called the discriminant), and calculate the two possible values of x.
If the discriminant is positive, there are two real roots. If the discriminant is zero, there is one real root. If the discriminant is negative, there are no real roots.
Tips and tricks:
To avoid mistakes, write down each step clearly and check your calculations.
To simplify square roots, look for perfect squares that are factors of the number under the root. For example, √81 = √(9 × 9) = 9.
To check your answers, plug them back into the original equation and see if they make it true.
Problem 2
Problem statement: Find the vertex, axis of symmetry, y-intercept and x-intercepts of the function f(x) = x - 4x - 5.
Answer: The vertex is (2,-9), the axis of symmetry is x = 2, the y-intercept is (0,-5), and the x-intercepts are (-1,0) and (5,0).
Steps:
Use the vertex form of a quadratic function to find the vertex and the axis of symmetry. Recall that the vertex form is f(x) = a(x - h) + k, where a, h and k are constants and a ≠ 0. The vertex is (h,k) and the axis of symmetry is x = h.
To convert the standard form of a quadratic function (f(x) = ax + bx + c) to the vertex form, you need to complete the square. For example, x - 4x - 5 = (x - 4x + 4) - 5 - 4 = (x - 2) - 9.
In this case, a = 1, h = 2, and k = -9. Therefore, the vertex is (2,-9) and the axis of symmetry is x = 2.
To find the y-intercept, plug in x = 0 into the function and solve for y. For example, f(0) = (0) - 4(0) - 5 = -5. Therefore, the y-intercept is (0,-5).
To find the x-intercepts, plug in y = 0 into the function and solve for x. You can use either the standard form or the vertex form. For example, using the standard form, you get x - 4x - 5 = 0. You can use the quadratic formula or factoring to solve this equation. In this case, factoring gives you (x + 1)(x - 5) = 0. Therefore, the x-intercepts are (-1,0) and (5,0).
Key concepts and formulas:
A quadratic function is a function of the form f(x) = ax + bx + c, where a, b and c are constants and a ≠ 0.
The standard form of a quadratic function tells you the y-intercept of the function (c) and whether it opens up (a > 0) or down (a < 0). To find the x-intercepts, you need to solve f(x) = 0.
The vertex form of a quadratic function tells you the vertex of the function (<
Problem 4
Problem statement: A ball is thrown upward from the top of a building with an initial velocity of 20 m/s. The height of the ball after t seconds is given by the function h(t) = -5t + 20t + 50, where h is in meters. How long does it take for the ball to reach its maximum height? What is the maximum height?
Answer: It takes 2 seconds for the ball to reach its maximum height. The maximum height is 70 meters.
Steps:
Use the vertex form of a quadratic function to find the vertex and the axis of symmetry. Recall that the vertex form is f(x) = a(x - h) + k, where a, h and k are constants and a ≠ 0. The vertex is (h,k) and the axis of symmetry is x = h.
To convert the standard form of a quadratic function (f(x) = ax + bx + c) to the vertex form, you need to complete the square. For example, -5t + 20t + 50 = -5(t - 4t) + 50 = -5(t - 4t + 4) + 50 + 20 = -5(t - 2) + 70.
In this case, a = -5, h = 2, and k = 70. Therefore, the vertex is (2,70) and the axis of symmetry is t = 2.
The vertex represents the maximum height of the ball, since the parabola opens down (a < 0). The axis of symmetry represents the time when the ball reaches its maximum height.
Therefore, it takes 2 seconds for the ball to reach its maximum height, and the maximum height is 70 meters.
Key concepts and formulas:
A quadratic function is a function of the form f(x) = ax + bx + c, where a, b and c are constants and a ≠ 0.
The standard form of a quadratic function tells you the y-intercept of the function (<
Conclusion
In this article, we have shown you how to solve some of the problems on page 132 of Pegi 11, a math textbook for high school students in Albania. We have explained the steps, concepts, formulas, and tips for each problem, and provided you with the answers. We hope that this guide has helped you understand and master the topics of quadratic functions and equations.
If you want to learn more about math or practice your skills, you can visit our website or follow us on social media. We have more articles, videos, quizzes, and exercises for you to explore. You can also contact us if you have any questions or feedback. Thank you for reading and happy learning! 4aad9cdaf3