Day 6 examples u1w14
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Graphs of absolute value functions are created by reflecting any portion of the original graph below the x-axis to above the x-axis; the example functions y = |- x - 2| and y = - |x - 2| + 1 demonstrate how absolute value graphs are constructed through reflection across the x-axis.
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Day 5 examples u1f13
Graphs of absolute value functions are created by reflecting any portion of the original graph below the x-axis to above the x-axis. Some examples of absolute value functions include y = |3x - 2|, y = |- x - 2|, and y = 2|x + 1|. More complex absolute value functions can also be graphed, such as y = - |x - 2| + 1.
Day 6 examples
Graphs of absolute value functions are created by reflecting any portion of the original graph below the x-axis to above the x-axis. The examples given are the absolute value functions y = |3x - 2|, y = |- x - 2|, y = 2|x + 1|, and y = - |x - 2| + 1 to demonstrate how to graph different absolute value expressions. The document assigns Ex 11 problems 1 to 6 and 8 as homework.
Lesson 3.4
This document discusses transformations of graphs, including shifting graphs horizontally and vertically, changing the amplitude to model changes in loudness, compressing graphs to fit standard view windows, and flipping graphs across the x-axis or y-axis. Some key points:
- Shifting a graph horizontally is done by adding or subtracting a value from the input, like f(x-2) or f(x)+3.
- Shifting vertically is done by adding or subtracting from the entire function, like f(x)+2.
- Increasing amplitude stretches a graph vertically, modeling increased loudness. Decreasing amplitude compresses vertically.
- Compressing a graph fits it within the standard view by multiplying the
Graphing Absolute Value
The document discusses graphing absolute value functions. It notes that absolute value functions always form a V-shape graph. It provides the general form of an absolute value function as y = a |x - h| + k, where (h, k) are the coordinates of the vertex and a describes the slope. It explains that horizontal shifts are performed by adding or subtracting h from x and vertical shifts add or subtract k from the function. Reflecting the graph over the x-axis flips it vertically. Examples demonstrate shifting graphs left, right, up, down and reflecting over the x-axis.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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An asymptote is an imaginary line that a graph approaches but never touches as it goes to infinity. The document provides examples of graphs with rational functions that have vertical and horizontal asymptotes. It also discusses the square root function and how it is the inverse of the quadratic function. The square root function is only defined for non-negative inputs. Changing the parameters a and b in the square root function changes the shape and orientation of the graph.
Transformations
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
Function transformations
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
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Graphs of absolute value functions are created by reflecting any portion of the original graph below the x-axis to above the x-axis. Some examples of absolute value functions include y = |3x - 2|, y = |- x - 2|, and y = 2|x + 1|. More complex absolute value functions can also be graphed, such as y = - |x - 2| + 1.
Day 6 examples
Graphs of absolute value functions are created by reflecting any portion of the original graph below the x-axis to above the x-axis. The examples given are the absolute value functions y = |3x - 2|, y = |- x - 2|, y = 2|x + 1|, and y = - |x - 2| + 1 to demonstrate how to graph different absolute value expressions. The document assigns Ex 11 problems 1 to 6 and 8 as homework.
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This document discusses transformations of graphs, including shifting graphs horizontally and vertically, changing the amplitude to model changes in loudness, compressing graphs to fit standard view windows, and flipping graphs across the x-axis or y-axis. Some key points:
- Shifting a graph horizontally is done by adding or subtracting a value from the input, like f(x-2) or f(x)+3.
- Shifting vertically is done by adding or subtracting from the entire function, like f(x)+2.
- Increasing amplitude stretches a graph vertically, modeling increased loudness. Decreasing amplitude compresses vertically.
- Compressing a graph fits it within the standard view by multiplying the
Graphing Absolute Value
The document discusses graphing absolute value functions. It notes that absolute value functions always form a V-shape graph. It provides the general form of an absolute value function as y = a |x - h| + k, where (h, k) are the coordinates of the vertex and a describes the slope. It explains that horizontal shifts are performed by adding or subtracting h from x and vertical shifts add or subtract k from the function. Reflecting the graph over the x-axis flips it vertically. Examples demonstrate shifting graphs left, right, up, down and reflecting over the x-axis.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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An asymptote is an imaginary line that a graph approaches but never touches as it goes to infinity. The document provides examples of graphs with rational functions that have vertical and horizontal asymptotes. It also discusses the square root function and how it is the inverse of the quadratic function. The square root function is only defined for non-negative inputs. Changing the parameters a and b in the square root function changes the shape and orientation of the graph.
Transformations
The document discusses how to graph functions by applying transformations to basic graphs. These transformations include vertical and horizontal shifts which move the graph up/down or left/right, reflections which flip the graph across an axis, and stretching or shrinking which make the graph taller or narrower. By identifying the transformations in an equation, one can graph it by applying the corresponding operations to the original function graph.
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The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
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1) Meru refers to the brain or vertebral column, which supports consciousness and life.
2) Meru-mani refers to a large bead used in rosaries for counting repetitions of a mantra during namasmaran (remembrance of God's name).
3) It is recommended not to pass over the meru-mani bead when counting repetitions, in order to replicate the flow of impulses between the left and right hemispheres of the brain during meditation and prayer.
Namasmaran And Romance Dr Shriniwas Janardan Kashalikar
The document discusses the concept of Namasmaran and how it relates to everyday life and romance. It was initially thought that Namasmaran was something only for those who had resigned from active life, like ascetics or retired people. However, the document argues that Namasmaran is actually a process of conscious evolution that makes one more objective, considerate, and able to react understandingly by reducing impulsiveness. In this way, Namasmaran can play a pivotal role in enriching everyday life and blossoming and immortalizing romance throughout one's life.
T H E F E T T E R S A N D T H E F R E E D O M Dr
It explains that Namasmaran helps to reconnect one's true self by focusing individual consciousness on a symbol of the cosmic consciousness. This practice can help overcome mediocrity in society and rise above worldly concerns. Namasmaran is described as the ultimate form of meditation and yoga, as it unifies all other spiritual practices. While difficult to describe fully in words, regular practice of Namasmaran can provide profound experiences of bliss.
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