Day 1 examples u1f13
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This document discusses horizontal shifts of functions. It states that a shift of "h" units to the right is represented by (x - h) and a shift of "h" units to the left is represented by (x + h). It provides examples of sketching graphs after a horizontal shift and finding the domain and range of a shifted function. The document also mentions stretching and compressing graphs vertically with "a" and gives examples of writing equations of translated functions in terms of a base function M(x). It concludes by assigning practice problems from page 168.
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Day 1 examples u1f13
This document discusses horizontal shifts of functions. It states that a shift of "h" units to the right is represented by (x - h) and a shift of "h" units to the left is represented by (x + h). It provides examples of sketching graphs after a horizontal shift and finding the domain and range of a shifted function. The document also mentions stretching and compressing graphs vertically with "a" and gives examples of writing equations of translated functions in terms of a base function M(x). It concludes by assigning practice problems from page 168.
Day 1 examples 2
This document discusses horizontal shifts of functions. It explains that a term like (x - h) shifts a function f(x) right by h units, while (x + h) shifts it left by h units. It provides examples of sketching graphs after horizontal shifts and finding domains and ranges of shifted functions. The document also covers describing shifted functions in words and finding equations of translated absolute value graphs.
Algebra 2 Section 2-6
This document defines key terms related to transformations of functions such as parent graph, parent function, translation, reflection, and dilation. It provides examples of how translations, reflections, and dilations affect the graph of a parent function. Translations move the graph horizontally or vertically by adding or subtracting a value from the input or output. Reflections flip the graph across an axis. Dilations compress or stretch the graph by multiplying the input or output by a value. The document includes examples showing how to describe transformations and graph the resulting function compared to the parent graph.
Transformations of functions
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
Horizontal Shifts of Quadratic Functions
- The graph of y = (x - h)2 has a vertex at (h,0) that is translated horizontally by h units from the origin.
- If h is positive, the translation is to the right, and if h is negative, the translation is to the left.
- Other quadratic graphs can be obtained by translating the basic graph of y = x2.
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 describes various transformations that can be applied to graphs of functions, including: vertical and horizontal translations that shift the graph along the y- or x-axis respectively; vertical and horizontal stretches/compressions that enlarge or reduce the graph along the y- or x-axis by a scale factor; and reflections about the x- or y-axis that flip the graph across that axis. Each transformation is defined symbolically by an algebraic modification to the original function, and its geometric effect on changing the x- and y-coordinates of points on the graph is also provided symbolically.
2.6.1 graphing techniques translations
Vertical translations move the graph up or down by adding or subtracting a number from the y-values. Horizontal translations move the graph left or right by adding or subtracting a number from the x-values. Reflections about the x or y-axis flip the graph across that axis by multiplying the x or y-values by -1. Stretching or compressing vertically or horizontally multiplies the y or x-values by a number, scaling the graph proportionally larger or smaller.
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Day 1 examples u1f13
This document discusses horizontal shifts of functions. It states that a shift of "h" units to the right is represented by (x - h) and a shift of "h" units to the left is represented by (x + h). It provides examples of sketching graphs after a horizontal shift and finding the domain and range of a shifted function. The document also mentions stretching and compressing graphs vertically with "a" and gives examples of writing equations of translated functions in terms of a base function M(x). It concludes by assigning practice problems from page 168.
Day 1 examples 2
This document discusses horizontal shifts of functions. It explains that a term like (x - h) shifts a function f(x) right by h units, while (x + h) shifts it left by h units. It provides examples of sketching graphs after horizontal shifts and finding domains and ranges of shifted functions. The document also covers describing shifted functions in words and finding equations of translated absolute value graphs.
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This document defines key terms related to transformations of functions such as parent graph, parent function, translation, reflection, and dilation. It provides examples of how translations, reflections, and dilations affect the graph of a parent function. Translations move the graph horizontally or vertically by adding or subtracting a value from the input or output. Reflections flip the graph across an axis. Dilations compress or stretch the graph by multiplying the input or output by a value. The document includes examples showing how to describe transformations and graph the resulting function compared to the parent graph.
Transformations of functions
The document discusses function transformations including shifts, reflections, and stretches/compressions. It defines these transformations and provides examples of how they affect the graph of a function. Specifically, it explains that a shift moves a graph up/down or left/right along an axis, a reflection flips the graph across an axis, and a stretch or compression changes the scale of the graph along an axis. Examples are given of reflecting across the x-axis or y-axis and horizontally or vertically stretching/compressing a function. In the end, students are asked to write equations for specific transformations of a quadratic function.
Horizontal Shifts of Quadratic Functions
- The graph of y = (x - h)2 has a vertex at (h,0) that is translated horizontally by h units from the origin.
- If h is positive, the translation is to the right, and if h is negative, the translation is to the left.
- Other quadratic graphs can be obtained by translating the basic graph of y = x2.
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.
Transformations
The document describes various transformations that can be applied to graphs of functions, including: vertical and horizontal translations that shift the graph along the y- or x-axis respectively; vertical and horizontal stretches/compressions that enlarge or reduce the graph along the y- or x-axis by a scale factor; and reflections about the x- or y-axis that flip the graph across that axis. Each transformation is defined symbolically by an algebraic modification to the original function, and its geometric effect on changing the x- and y-coordinates of points on the graph is also provided symbolically.
2.6.1 graphing techniques translations
Vertical translations move the graph up or down by adding or subtracting a number from the y-values. Horizontal translations move the graph left or right by adding or subtracting a number from the x-values. Reflections about the x or y-axis flip the graph across that axis by multiplying the x or y-values by -1. Stretching or compressing vertically or horizontally multiplies the y or x-values by a number, scaling the graph proportionally larger or smaller.
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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This document discusses transformations of graphs including translations, stretches, and reflections. It defines vertical and horizontal translations as shifting the graph of a function up or down by adding a constant c, or shifting it left or right by changing the input by a constant b. Stretches are defined as vertically stretching by a scale factor a or horizontally by 1/a. Reflections are defined as reflecting the graph across the x or y-axis. Examples and exercises are provided to illustrate each type of transformation.
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The document discusses various ways to transform graphs of functions including vertical and horizontal shifts, reflections, stretching and shrinking. It provides examples of transforming graphs by applying these techniques sequentially in a specified order: first horizontal shifting, then stretching or shrinking, then reflecting, and finally vertical shifting. The summary provides examples of transforming an original function f(x) into another function g(x) by applying these transformations.
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The document discusses reflections of graphs over the x-axis and y-axis. It provides examples of graphs with given x-intercepts and y-intercepts, and asks to determine the x-intercepts and y-intercepts of the reflected graphs over the x-axis and y-axis. It also gives the domain and range of a function f(x) and asks to determine the domain and range when the function is reflected as -f(x) and f(-x). The document provides practice problems from an assignment on pages 183 problems 1 through 11 and problem 13.
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Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
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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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Day 1 examples u1f13
- 1. Intro to Translation Horizontal Shifts Recall from 30SP "k" shifts f(x) vertically up/down by k-units "h" shifts f(x) horizontally left/right (x - h) --> shifts right by h-units (x + h) --> shifts left by h-units "a" stretches or compresses f(x) vertically a>1 vertical stretch occurs a<1 vertical compress occurs
- 2. Need to know some base graphs:
- 4. ex. Given the graph of f(x), Sketch the graph of y = f(x -1) - 3 ex. The function y = f(x) has the domain -1 x 4 and range 2 y 7. What are the domain and range of y = f(x - 5) + 1?≤ ≤ ≤ ≤
- 5. State the equation for the following functions in relation to M(x) ex. Describe the following in words: ex. The graph of y = |x| is translated 7 units right and 6 units up. What is the equation of the image graph? Assignment p.168 #1, 2, 4-16