Absolute Value Functions & Graphs - Module 4 and 5
•Download as PPT, PDF•
2 likes•4,579 views
1) Absolute value functions have a V-shaped graph called the parent function y=|x|. 2) Absolute value functions can be graphed by identifying the vertex (x,y) coordinates using the equation, choosing x-values to plot points, and connecting the points. 3) Graphing calculators can also be used to graph absolute value functions by entering the equation in y=, using math mode to input the absolute value symbol, and graphing the function.
Report
Share
Report
Share
![Absolute Value Functions ,[object Object],[object Object],[object Object],[object Object],[object Object]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Absolute Value Functions & Graphs
Absolute value functions have a V-shaped graph. There are three main ways to graph an absolute value function: using a table of values, using a graphing calculator, or interpreting the equation. The vertex of an absolute value graph is always the x-coordinate inside the absolute value signs. To graph, one finds the vertex and then plots additional points on either side, connecting them to form the V-shape. The number outside the absolute value signs moves the graph up or down, while the number inside moves it left or right.
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.
Absolute value functions
The document discusses absolute value functions. It defines absolute value and notes that the graph of an absolute value function is V-shaped and symmetric about the y-axis. It provides details on how to graph an absolute value function given its equation, including identifying the vertex and using the slope and symmetry to plot additional points and complete the graph.
Two step equations
- Two-step equations can be solved using a do/undo box method where the operations on the "do" side represent the mathematical operations relating to the variable, and the "undo" side represents the inverse operations in reverse order to solve for the variable.
- The document provides examples of setting up two-step equations like 2x = -2 and -3y - 5 = -20 in a do/undo box, with the "do" side showing the operations on the same side as the variable and the "undo" side showing the inverse operations to solve step-by-step for the variable.
- Using this method of a do/undo box with the operations written in order on one side
2.8 Absolute Value Functions
Absolute value functions have a V-shape and model situations involving distance or edges. The graph can be transformed by changing the slope (a), shifting the vertex horizontally (h), or shifting the vertex vertically (k). To graph, identify the vertex and axis of symmetry, then use the slope to sketch the right side and symmetry to complete the left. Writing the equation involves identifying the vertex (h, k) and slope (a).
Antiderivatives And Slope Fields
The document discusses slope fields and how they can be used to graphically represent solutions to differential equations. It provides examples of drawing slope fields by hand and using a graphing calculator. Initial value problems are defined as differential equations where an initial condition is given to determine the constant term. The document also discusses the differences between definite and indefinite integrals, and provides tips for using a graphing calculator to find and graph indefinite integrals.
Absolute value functions
1. Absolute-value functions can be transformed through translations, reflections, stretches, and compressions. Translations move the graph up/down or left/right and change the vertex. Reflections flip the graph across an axis. Stretches and compressions make the graph wider/narrower or taller/shorter.
2. Examples show how to write the rule for a transformed absolute-value function g(x) based on an original function f(x). The transformations can be vertical/horizontal shifts, reflections, or vertical/horizontal stretches/compressions.
3. Graphs confirm that the transformations are applied as expected to f(x), resulting in the graph of g(x) with the
8.1 exponential growth
This document defines and explains exponential functions, which are functions that include the expression bx where b is a positive number other than 1. It discusses the shape of exponential graphs, asymptotes, graphing the general forms y=abx and y=a(b^x)+h+k, exponential growth models, and compound interest formulas. Examples are provided to illustrate key concepts.
Recommended
Absolute Value Functions & Graphs
Absolute value functions have a V-shaped graph. There are three main ways to graph an absolute value function: using a table of values, using a graphing calculator, or interpreting the equation. The vertex of an absolute value graph is always the x-coordinate inside the absolute value signs. To graph, one finds the vertex and then plots additional points on either side, connecting them to form the V-shape. The number outside the absolute value signs moves the graph up or down, while the number inside moves it left or right.
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.
Absolute value functions
The document discusses absolute value functions. It defines absolute value and notes that the graph of an absolute value function is V-shaped and symmetric about the y-axis. It provides details on how to graph an absolute value function given its equation, including identifying the vertex and using the slope and symmetry to plot additional points and complete the graph.
Two step equations
- Two-step equations can be solved using a do/undo box method where the operations on the "do" side represent the mathematical operations relating to the variable, and the "undo" side represents the inverse operations in reverse order to solve for the variable.
- The document provides examples of setting up two-step equations like 2x = -2 and -3y - 5 = -20 in a do/undo box, with the "do" side showing the operations on the same side as the variable and the "undo" side showing the inverse operations to solve step-by-step for the variable.
- Using this method of a do/undo box with the operations written in order on one side
2.8 Absolute Value Functions
Absolute value functions have a V-shape and model situations involving distance or edges. The graph can be transformed by changing the slope (a), shifting the vertex horizontally (h), or shifting the vertex vertically (k). To graph, identify the vertex and axis of symmetry, then use the slope to sketch the right side and symmetry to complete the left. Writing the equation involves identifying the vertex (h, k) and slope (a).
Antiderivatives And Slope Fields
The document discusses slope fields and how they can be used to graphically represent solutions to differential equations. It provides examples of drawing slope fields by hand and using a graphing calculator. Initial value problems are defined as differential equations where an initial condition is given to determine the constant term. The document also discusses the differences between definite and indefinite integrals, and provides tips for using a graphing calculator to find and graph indefinite integrals.
Absolute value functions
1. Absolute-value functions can be transformed through translations, reflections, stretches, and compressions. Translations move the graph up/down or left/right and change the vertex. Reflections flip the graph across an axis. Stretches and compressions make the graph wider/narrower or taller/shorter.
2. Examples show how to write the rule for a transformed absolute-value function g(x) based on an original function f(x). The transformations can be vertical/horizontal shifts, reflections, or vertical/horizontal stretches/compressions.
3. Graphs confirm that the transformations are applied as expected to f(x), resulting in the graph of g(x) with the
8.1 exponential growth
This document defines and explains exponential functions, which are functions that include the expression bx where b is a positive number other than 1. It discusses the shape of exponential graphs, asymptotes, graphing the general forms y=abx and y=a(b^x)+h+k, exponential growth models, and compound interest formulas. Examples are provided to illustrate key concepts.
3.3 graphs of exponential functions
This document discusses exponential functions including:
- Exponential growth graphs move away from the x-axis quickly when b > 1, while decay graphs move towards it when 0 < b < 1. Both have a y-intercept of (0, a).
- Functions of the form y = a(b)x have a horizontal asymptote at y = 0.
- The domain is the set of all input values x, while the range is the set of all output values y.
- Examples are provided of identifying growth/decay, domain, range, asymptotes and y-intercepts of exponential functions. Graphing techniques including making tables and connecting points with smooth curves are also outlined.
Graphing a line using Slope-Intercept form
This document explains how to graph a line using the slope-intercept form of a line, y=mx+b, where m is the slope and b is the y-intercept. It gives the example of the line with slope -2/3 and y-intercept of 6, so the equation is y=-2/3x+6. To graph this line, you first plot the y-intercept point (0,6), then find a second point by moving down 2 units and right 3 units from the first point and connect the two points with a straight line with arrows on the ends.
Coordinate plane
The document introduces the coordinate plane as a tool for working with equations in two variables. It explains how to create a coordinate plane by drawing perpendicular x and y axes that intersect at the origin, and using the axes to locate points on the plane using ordered pairs of numbers (x,y) as coordinates. It describes how to plot points by starting at the origin and counting along the x-axis and y-axis according to the x and y coordinates of the point. Finally, it divides the coordinate plane into four quadrants.
Quadratic graphs- features
In this presentation, you will see examples of finding the image when given an element of the domain and finding an element of the domain for a given image.
Graphing rational functions
This document discusses how to graph rational functions. It explains that rational functions have vertical and horizontal asymptotes. Vertical asymptotes occur when the denominator is set equal to zero. Horizontal asymptotes depend on the degrees of the numerator and denominator. If the denominator degree is greater, the horizontal asymptote is the x-axis. If the degrees are the same, the asymptote is found by dividing the leading coefficients. If there is a vertical shift, the asymptote is the same as the shift value. Examples are provided to demonstrate graphing different rational functions based on their asymptotes.
Grafica de Funciones Trigonométricas.
Grafica de Funciones Trigonométricas con sus características:
1. Seno.
2. Coseno.
3. Análisis de gráficas: traslación, reflexión, compresión y alargamiento.
4. Tangente.
5. Cotangente.
6. Secante.
7. Cosecante.
3
This document discusses rational functions and their graphs. It covers key concepts such as finding and graphing vertical and horizontal asymptotes of rational functions. Examples are provided to illustrate how to find asymptotes and graph different types of rational functions, including transformations and slant asymptotes. Applications involving rational functions are also presented.
Rational functions 13.1 13.2
This document provides an overview of rational functions and how to graph them. It discusses identifying vertical and horizontal asymptotes by analyzing the degrees of the numerator and denominator polynomials. It gives examples of multiplying, dividing, and graphing rational functions. Students are asked to find asymptotes, state domains and ranges, and graph rational functions. They are also given examples and problems for multiplying and dividing rational expressions.
A2 1 linear fxns
This document discusses linear functions and how to graph them. It defines relations and functions, noting that a function has each input mapped to only one output. It states that a linear function forms a straight line on a graph. Methods for graphing linear functions include creating a table of x- and y-values by plugging inputs into the linear equation and plotting the points, or finding the x- and y-intercepts where the line crosses the axes.
Rational, Irrational & Absolute Value Functions Review
The document discusses various topics related to graphing rational, irrational, and absolute value functions. It provides instructions on how to graph these functions by finding intercepts, asymptotes, points of discontinuity, and applying transformations. It also discusses how to simplify and solve rational expressions and inequalities.
9.2 graphing simple rational functions
This document provides instructions for graphing rational functions without making a table. It explains that the graph is a hyperbola made of two symmetrical branches. It outlines steps to sketch the asymptotes and plot points to draw the branches, being careful not to cross the asymptotes. The steps are to sketch the asymptotes, choose x-values on each side to plug into the function, plot the points, and draw the branches without crossing the asymptotes.
Application of derivatives
The document describes a problem where a frog jumps up 3 steps per day and down 2 steps the next day. It asks how many days it will take the frog to reach the top of a 6 step stair. It provides the key information that the frog jumps up 3 steps one day and down 2 the next day, alternating in this pattern until it reaches the top of a 6 step stair.
Section 4 2 math
Solve addition and subtraction equations. The opposite of addition is subtraction and the opposite of multiplication is division. Whatever is done to one side of an equation must also be done to the other side to maintain equality. Examples are provided of solving different types of one-step equations involving addition, subtraction, multiplication and division.
A1 1 linear fxns
Linear functions create a straight line on a graph where each point (x, y) that satisfies the linear equation lies. The document outlines how to graph linear functions by creating a table of x-values to plug into the linear equation to find the corresponding y-values, plotting these points and connecting them with a straight line.
11.6 graphing linear inequalities in two variables
This document discusses how to graph linear inequalities in two variables. It provides examples of graphing various inequalities, including 2x + 3y ≤ 6, 2x + y > 3, and y < 3x. The process involves graphing the boundary line defined by the equal case of the inequality and then shading the appropriate region based on checking if a test point satisfies the inequality. Key steps are to draw the boundary line as solid or dashed based on the symbols (< or > versus ≤ or ≥) and then shade the region containing points that satisfy the inequality.
5.4 Slope Intercept Form
The document provides an overview of slope-intercept form for linear equations. It defines slope as the constant rate of change and the y-intercept as the y-value when x=0. Slope-intercept form is given as y=mx+b, where m is the slope and b is the y-intercept. This form can be used to graph lines, find points, and describe situations with a constant rate of change. Understanding slope-intercept form is important for showing what y-values occur at different x-values for a linear relationship.
6.6 Graphing Inequalities In Two Variables
This document discusses graphing linear inequalities in two variables. It provides definitions of key terms like half-plane and boundary. It also gives helpful hints for graphing different types of inequalities based on whether the sign is >, <, ≥, or ≤ and whether it involves just x and y or a slanted line. Examples are provided to demonstrate how to graph inequalities like y > 3, x - 2, y - 3x + 2, and an applied problem involving nickels and dimes. The key steps for graphing a linear inequality on the coordinate plane are outlined.
Rational Functions
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
M17 t1 notes
The document provides step-by-step instructions for graphing rational functions. It explains that rational functions are fractions of polynomials, and defines key features like domains, zeros, vertical and horizontal asymptotes. Examples are given to demonstrate identifying these features and graphing the rational function based on them. Steps include finding intercepts, asymptotes, determining where the graph lies relative to the x-axis, and combining these elements to graph the full function.
Final review
Limits define the highest or lowest value that a function can approach as the input value approaches a specific number. There are left-hand limits and right-hand limits, depending on whether the input is approaching the number from the left or right side. The limit may not exist if there is a jump discontinuity, infinite discontinuity, or discontinuity over an interval. To find the limit, one can use substitution by substituting the specific number directly into the function. Alternatively, simplification can be used if substitution does not work, such as by factoring the top quantity to cancel out terms. If those fail, a table of values or graphing the function can help determine the limit.
S7 investigating vectors student worksheet (colour)
This document contains a student worksheet for investigating vectors in 2D and 3D using Autograph software. It includes tasks on expressing vectors in terms of other vectors, finding vector equations of lines given conditions, calculating lengths and angles of vectors, and using vectors to solve geometric problems in 2D and 3D. The worksheet guides students through using Autograph's vector tools to explore and check their work.
Unit+7 1
This document provides instruction on graphing quadratic functions. It defines key terms like parabola, vertex, axis of symmetry, and solutions. It explains that the vertex is the highest or lowest point and its x-coordinate is the axis of symmetry. The solutions are the x-values where the parabola crosses the x-axis. Examples show how to identify the vertex and whether the function has a maximum or minimum. The last example demonstrates graphing a quadratic function using a table of values.
More Related Content
What's hot
3.3 graphs of exponential functions
This document discusses exponential functions including:
- Exponential growth graphs move away from the x-axis quickly when b > 1, while decay graphs move towards it when 0 < b < 1. Both have a y-intercept of (0, a).
- Functions of the form y = a(b)x have a horizontal asymptote at y = 0.
- The domain is the set of all input values x, while the range is the set of all output values y.
- Examples are provided of identifying growth/decay, domain, range, asymptotes and y-intercepts of exponential functions. Graphing techniques including making tables and connecting points with smooth curves are also outlined.
Graphing a line using Slope-Intercept form
This document explains how to graph a line using the slope-intercept form of a line, y=mx+b, where m is the slope and b is the y-intercept. It gives the example of the line with slope -2/3 and y-intercept of 6, so the equation is y=-2/3x+6. To graph this line, you first plot the y-intercept point (0,6), then find a second point by moving down 2 units and right 3 units from the first point and connect the two points with a straight line with arrows on the ends.
Coordinate plane
The document introduces the coordinate plane as a tool for working with equations in two variables. It explains how to create a coordinate plane by drawing perpendicular x and y axes that intersect at the origin, and using the axes to locate points on the plane using ordered pairs of numbers (x,y) as coordinates. It describes how to plot points by starting at the origin and counting along the x-axis and y-axis according to the x and y coordinates of the point. Finally, it divides the coordinate plane into four quadrants.
Quadratic graphs- features
In this presentation, you will see examples of finding the image when given an element of the domain and finding an element of the domain for a given image.
Graphing rational functions
This document discusses how to graph rational functions. It explains that rational functions have vertical and horizontal asymptotes. Vertical asymptotes occur when the denominator is set equal to zero. Horizontal asymptotes depend on the degrees of the numerator and denominator. If the denominator degree is greater, the horizontal asymptote is the x-axis. If the degrees are the same, the asymptote is found by dividing the leading coefficients. If there is a vertical shift, the asymptote is the same as the shift value. Examples are provided to demonstrate graphing different rational functions based on their asymptotes.
Grafica de Funciones Trigonométricas.
Grafica de Funciones Trigonométricas con sus características:
1. Seno.
2. Coseno.
3. Análisis de gráficas: traslación, reflexión, compresión y alargamiento.
4. Tangente.
5. Cotangente.
6. Secante.
7. Cosecante.
3
This document discusses rational functions and their graphs. It covers key concepts such as finding and graphing vertical and horizontal asymptotes of rational functions. Examples are provided to illustrate how to find asymptotes and graph different types of rational functions, including transformations and slant asymptotes. Applications involving rational functions are also presented.
Rational functions 13.1 13.2
This document provides an overview of rational functions and how to graph them. It discusses identifying vertical and horizontal asymptotes by analyzing the degrees of the numerator and denominator polynomials. It gives examples of multiplying, dividing, and graphing rational functions. Students are asked to find asymptotes, state domains and ranges, and graph rational functions. They are also given examples and problems for multiplying and dividing rational expressions.
A2 1 linear fxns
This document discusses linear functions and how to graph them. It defines relations and functions, noting that a function has each input mapped to only one output. It states that a linear function forms a straight line on a graph. Methods for graphing linear functions include creating a table of x- and y-values by plugging inputs into the linear equation and plotting the points, or finding the x- and y-intercepts where the line crosses the axes.
Rational, Irrational & Absolute Value Functions Review
The document discusses various topics related to graphing rational, irrational, and absolute value functions. It provides instructions on how to graph these functions by finding intercepts, asymptotes, points of discontinuity, and applying transformations. It also discusses how to simplify and solve rational expressions and inequalities.
9.2 graphing simple rational functions
This document provides instructions for graphing rational functions without making a table. It explains that the graph is a hyperbola made of two symmetrical branches. It outlines steps to sketch the asymptotes and plot points to draw the branches, being careful not to cross the asymptotes. The steps are to sketch the asymptotes, choose x-values on each side to plug into the function, plot the points, and draw the branches without crossing the asymptotes.
Application of derivatives
The document describes a problem where a frog jumps up 3 steps per day and down 2 steps the next day. It asks how many days it will take the frog to reach the top of a 6 step stair. It provides the key information that the frog jumps up 3 steps one day and down 2 the next day, alternating in this pattern until it reaches the top of a 6 step stair.
Section 4 2 math
Solve addition and subtraction equations. The opposite of addition is subtraction and the opposite of multiplication is division. Whatever is done to one side of an equation must also be done to the other side to maintain equality. Examples are provided of solving different types of one-step equations involving addition, subtraction, multiplication and division.
A1 1 linear fxns
Linear functions create a straight line on a graph where each point (x, y) that satisfies the linear equation lies. The document outlines how to graph linear functions by creating a table of x-values to plug into the linear equation to find the corresponding y-values, plotting these points and connecting them with a straight line.
11.6 graphing linear inequalities in two variables
This document discusses how to graph linear inequalities in two variables. It provides examples of graphing various inequalities, including 2x + 3y ≤ 6, 2x + y > 3, and y < 3x. The process involves graphing the boundary line defined by the equal case of the inequality and then shading the appropriate region based on checking if a test point satisfies the inequality. Key steps are to draw the boundary line as solid or dashed based on the symbols (< or > versus ≤ or ≥) and then shade the region containing points that satisfy the inequality.
5.4 Slope Intercept Form
The document provides an overview of slope-intercept form for linear equations. It defines slope as the constant rate of change and the y-intercept as the y-value when x=0. Slope-intercept form is given as y=mx+b, where m is the slope and b is the y-intercept. This form can be used to graph lines, find points, and describe situations with a constant rate of change. Understanding slope-intercept form is important for showing what y-values occur at different x-values for a linear relationship.
6.6 Graphing Inequalities In Two Variables
This document discusses graphing linear inequalities in two variables. It provides definitions of key terms like half-plane and boundary. It also gives helpful hints for graphing different types of inequalities based on whether the sign is >, <, ≥, or ≤ and whether it involves just x and y or a slanted line. Examples are provided to demonstrate how to graph inequalities like y > 3, x - 2, y - 3x + 2, and an applied problem involving nickels and dimes. The key steps for graphing a linear inequality on the coordinate plane are outlined.
Rational Functions
This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
M17 t1 notes
The document provides step-by-step instructions for graphing rational functions. It explains that rational functions are fractions of polynomials, and defines key features like domains, zeros, vertical and horizontal asymptotes. Examples are given to demonstrate identifying these features and graphing the rational function based on them. Steps include finding intercepts, asymptotes, determining where the graph lies relative to the x-axis, and combining these elements to graph the full function.
What's hot (19)
Rational, Irrational & Absolute Value Functions Review
Rational, Irrational & Absolute Value Functions Review
11.6 graphing linear inequalities in two variables
11.6 graphing linear inequalities in two variables
Similar to Absolute Value Functions & Graphs - Module 4 and 5
Final review
Limits define the highest or lowest value that a function can approach as the input value approaches a specific number. There are left-hand limits and right-hand limits, depending on whether the input is approaching the number from the left or right side. The limit may not exist if there is a jump discontinuity, infinite discontinuity, or discontinuity over an interval. To find the limit, one can use substitution by substituting the specific number directly into the function. Alternatively, simplification can be used if substitution does not work, such as by factoring the top quantity to cancel out terms. If those fail, a table of values or graphing the function can help determine the limit.
S7 investigating vectors student worksheet (colour)
This document contains a student worksheet for investigating vectors in 2D and 3D using Autograph software. It includes tasks on expressing vectors in terms of other vectors, finding vector equations of lines given conditions, calculating lengths and angles of vectors, and using vectors to solve geometric problems in 2D and 3D. The worksheet guides students through using Autograph's vector tools to explore and check their work.
Unit+7 1
This document provides instruction on graphing quadratic functions. It defines key terms like parabola, vertex, axis of symmetry, and solutions. It explains that the vertex is the highest or lowest point and its x-coordinate is the axis of symmetry. The solutions are the x-values where the parabola crosses the x-axis. Examples show how to identify the vertex and whether the function has a maximum or minimum. The last example demonstrates graphing a quadratic function using a table of values.
7-1 Exploring Quadratic Functions
This document discusses key concepts about quadratic functions including that they form parabolas with an axis of symmetry, how to identify the a, b, and c coefficients, and that the vertex represents the minimum or maximum value depending on whether a is positive or negative. It also explains how to graph quadratic functions using a table of values on a calculator or by hand and how the width of the parabola relates to the value of a.
Graphing rational functions
The document discusses graphs of rational functions and how to analyze them. It covers identifying vertical and horizontal asymptotes analytically by finding values where the denominator is zero or analyzing behavior as x approaches positive/negative infinity. Examples show graphing points around asymptotes and confirming trends to sketch the graph. Key steps are finding intercepts and asymptotes, making a T-chart of values, and graphing with curves between points and asymptotes. Practice problems are provided to apply these skills.
Graphing rational functions
The document discusses graphs of rational functions and how to analyze them. It covers identifying vertical and horizontal asymptotes analytically by finding values where the denominator is zero or analyzing behavior as x approaches positive/negative infinity. Examples show graphing rational functions by plotting points near asymptotes and connecting them, confirming asymptotes numerically. Key steps are finding intercepts and asymptotes, making a table near asymptotes, and graphing points and asymptotes. The reader is assigned practice problems applying these skills.
Graphing Linear Equations Lesson
There are four main methods to graph linear equations:
1) Point plotting involves choosing x-values, substituting them into the equation to find corresponding y-values, and plotting the points.
2) Using intercepts finds the x and y-intercepts by substituting 0 for x or y and solving for the other variable.
3) The slope-intercept form finds the slope and y-intercept to graph the line.
4) A graphing calculator can be used by inputting the equation in slope-intercept form (y=mx + b) and evaluating it to graph the line.
2021 noveno guia en casa matematicas periodo 3
The document provides information about mathematics courses and teachers for 9th grade students at Institución Educativa Inem "Jorge Isaacs" including:
- A list of the different mathematics and geometry courses/teachers for each of the 16 9th grade groups.
- Criteria for evaluating student activities and assignments including developing work individually, justifying work, submitting photo evidence of work, and meeting deadlines.
- Learning objectives, standards, and competencies related to real numbers, algebraic expressions, linear equations and systems of linear equations.
- An introduction to the third study guide which will cover linear equations of lines and quadratic functions including definitions, examples, and how to calculate elements of parabolas
Exploring Quadratic Functions
Quadratic functions form parabolas and have an axis of symmetry. They can be written in standard form as ax^2 + bx + c = y, where a cannot equal 0. The vertex of the parabola occurs at the minimum if a > 0 and it opens upwards, or at the maximum if a < 0 and it opens downwards. Quadratic functions can be graphed using a table of values in a graphing calculator or by hand. Ordering quadratic functions from widest to narrowest depends on the coefficient a, with larger absolute values of a producing narrower parabolas.
7 1 Exploring Quadratic Functions
Quadratic functions form parabolas and have an axis of symmetry. They can be written in standard form as ax^2 + bx + c = y, where a cannot equal 0. The vertex of the parabola occurs at the minimum if a > 0 and it opens upwards, or at the maximum if a < 0 and it opens downwards. Quadratic functions can be graphed using a table of values in a graphing calculator or by hand. Ordering quadratic functions from widest to narrowest depends on the coefficient a, with larger absolute values of a producing narrower parabolas.
Graphing Linear Equations Lesson
There are several methods for graphing linear equations:
1) Point plotting involves choosing x-values, substituting them into the equation to find corresponding y-values, and plotting the points on a graph.
2) Using intercepts involves finding the x-intercept where y=0 and the y-intercept where x=0 and graphing those points.
3) The slope-intercept form y=mx+b allows you to find the slope m and y-intercept b to graph the line.
4) A graphing calculator can also be used to graph linear equations by inputting them in slope-intercept form y=mx+b.
Linear Inequality slayerix
1) Solving linear inequalities involves using the same concepts as solving linear equations to isolate the variable, but inequalities have infinite solutions rather than a single solution.
2) Inequality signs (<, >, ≤, ≥) indicate whether values are less than, greater than, less than or equal to, or greater than or equal to another value.
3) Graphing linear inequalities on a number line or coordinate plane involves graphing the corresponding equality and then shading the appropriate side based on the inequality sign and whether it includes the endpoint or not.
Quadratic function in standard form (y = ax^2 +bx + c
Graphing quadratic functions, identify maximum and minimum values, how to identify the lines of symmetry, how to find the vertex,
Linear Functions Presentation
My final project that was to be a PowerPoint that could be worked on by individual students. This PowerPoint goes over linear functions.
Números Complejos.
Números Complejos:
1. Representación gráfica de los números complejos.
2. Conjugado y opuesto de un número complejo.
3. Operaciones con números complejos.
Graphing Quadradic
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of such a function is a parabola, and discusses key features of parabolas including whether they open up or down based on the sign of a, their line of symmetry, and how to find the vertex. The document gives step-by-step instructions for graphing a quadratic function in standard form, including finding the line of symmetry, locating the vertex, and using reflection across the line of symmetry to graph the full parabola.
Graphing Quadratics
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of a quadratic function is a parabola that can open up or down, with the lowest or highest point being the vertex. It describes how to find the line of symmetry, which is x = -b/2a and always passes through the vertex. The steps to graph a quadratic function are given as finding the line of symmetry, plugging it into the original equation to find the vertex coordinates, then finding two other points and reflecting them across the line of symmetry.
Graphing Quadratics
This document provides information about graphing quadratic functions in the form y = ax^2 + bx + c. It explains that the graph of such a function is a parabola, and discusses how to find the vertex and line of symmetry. The line of symmetry is given by x = -b/2a, and the vertex is found by plugging the x-value from the line of symmetry into the original equation to get the y-value. An example of graphing the function y = 2x^2 - 4x - 1 is presented to demonstrate these concepts.
April 4, 2014
Today's lesson reviews graphing quadratic functions by examining how the b value changes the parabola's position. The teacher discusses important concepts from the previous day like the axis of symmetry and vertex. Students learn that a positive b value results in a shift of the parabola to the left. For class work, students will practice graphing quadratic functions by finding the axis of symmetry and vertex, then plotting additional points to complete the parabola.
Similar to Absolute Value Functions & Graphs - Module 4 and 5 (20)
S7 investigating vectors student worksheet (colour)
S7 investigating vectors student worksheet (colour)
Quadratic function in standard form (y = ax^2 +bx + c
Quadratic function in standard form (y = ax^2 +bx + c
More from toni dimella
Parent functions and Transformations
This document discusses transformations of parent functions. It defines a parent function as the simplest form of a function, such as y=x, y=x^2, etc. Transformations include horizontal and vertical shifts which move the graph left, right, up or down; stretches which multiply the y-values making the graph skinnier; shrinks which reduce the y-values making the graph fatter; and reflections which flip the graph over the x-axis. Examples are provided to demonstrate how to graph different transformations of common parent functions. The document concludes by describing the transformation y=4x-2-5 as a shrink by a factor of 4, a right shift of 2 units, and a down shift of 5
Global Marketing in HE
Globalization has significantly impacted higher education by increasing opportunities for internationalization. Universities now recruit more international students, collaborate on research globally, and offer study abroad programs to give students international experience. This course will examine how higher education institutions can develop strategies to internationalize their campuses and curriculum to prepare students for success in a globalized world.
Module 2 lesson 4 notes
Parent functions are families of graphs that share unique properties. Transformations can move the graph around the plane. The main parent functions explored are the constant, linear, absolute value, quadratic, cubic, square root, cubic root, and exponential functions. Each has a characteristic shape and number of intercepts. Domains and ranges depend on the specific function but often extend to positive and negative infinity.
Module 2 Lesson 2 Notes
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
Module 1 Lesson 1 Remediation Notes
The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
M1 L5 Remediation Notes
This document discusses solving systems of equations and inequalities through three main methods: graphing, substitution, and elimination. It provides examples of each method. For graphing systems, it explains the three possibilities for the graphs: consistent systems with one solution where the lines intersect, inconsistent systems with no solution where the lines are parallel, and dependent systems with infinite solutions where the lines coincide. It then works through examples of using substitution and elimination to solve systems algebraically. [/SUMMARY]
Relations and Functions
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
Multiplying Polynomials (no conjugates)
This document discusses techniques for multiplying polynomials including FOIL, the double distributive property, and squaring binomials. It provides examples of squaring binomial expressions like (x + 3)2 by multiplying (x + 3) by itself using FOIL. The document also lists more example problems but does not show the solutions.
Fractions, Decimals, and Percents
This document defines decimals, fractions, and percents and provides steps for converting between them. Decimals are numbers with a decimal point, fractions show parts of a whole, and percents express amounts out of 100. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a fraction to a percent, change it to a decimal then multiply by 100. Converting between other forms follows similar steps of changing the number to an equivalent decimal or percent value.
C3 Study Slides - MAT 151
This document contains lecture slides about statistics for describing, exploring, and comparing data. It discusses measures of center such as the mean, median, and mode. It also discusses variance and standard deviation as measures of spread. Additional topics covered include finding the mode, determining if a value is unusually high or low based on the mean and standard deviation, calculating percentiles, and comparing the detail provided by different graphic displays of data.
C2 Study Slides - MAT 151
The document contains multiple choice questions about summarizing and graphing data. It asks questions about outliers, frequency distributions, class boundaries, histograms, and types of charts including pie charts. One question asks the reader to use a pie chart showing housing types to find the number of people living in single family housing in a town of 12,200 people.
C1 Study Slides - MAT151
The document contains lecture slides on introductory statistics topics including definitions of population, quantitative vs qualitative data, types of measurement scales, and examples of different sampling methods and study designs. Key points covered are the definition of a population as the complete collection of all elements, examples of quantitative data like weights vs qualitative nominal categories, ordinal scales involving ranking, and retrospective study designs using existing historical data.
C3 test Doc
This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalences, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
C3 test
This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalence, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
Intro to Logs
The document introduces logarithms, defining them as the exponent that a fixed number (the base) must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and covers the key properties of logarithms including product, quotient, power, expanding, condensing, and the change of base formula.
Logs
The document introduces logarithms, defining them as the exponent that a base number must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and discusses properties of logarithms such as the product, quotient, and power properties. It also covers expanding, condensing, and changing the base of logarithmic expressions.
Intro to Polynomials
This document discusses polynomial functions and how to graph them. It defines a polynomial as a sum of terms with non-negative integer exponents. Polynomial graphs are smooth curves that may be lines, parabolas, or higher-order curves. To graph a polynomial, one determines the end behavior from the leading term, finds the x-intercepts by setting the polynomial equal to 0, and uses intercepts and test points to plot the graph over intervals. Multiplicity of roots affects whether the graph crosses or is tangent to the x-axis at those points.
Completing the Square
The document explains how to solve quadratic equations by completing the square. It defines a perfect square trinomial as having the form x^2 + bx + c, where c is the square of half of b. It provides steps for completing the square, which involves adding a constant term to both sides of the equation such that the left side becomes a perfect square trinomial that can be factorized. This process results in the solution(s) to the quadratic equation. Two examples demonstrating this process are included.
Graph Quadratics
The document discusses graphing quadratic functions in standard form (y=ax^2 + bx + c). It explains that the graph is a parabola that can open up or down depending on whether a is positive or negative. The line of symmetry for the parabola passes through the vertex and is given by the equation x=-b/2a. The steps to graph are: 1) find the line of symmetry, 2) plug the x-value into the original equation to find the vertex, 3) find two other points and reflect them across the line of symmetry.
Composite functions
Composite functions are formed by taking the output of one function and using it as the input of another function. This is shown notationally as f(g(x)), where the result of g(x) is used as the input for f. Changing the order of the functions changes the result, as the output of the inner function determines the input to the outer function. Examples show evaluating composite functions by substituting the output of the inner function into the outer function and simplifying.
More from toni dimella (20)
Absolute Value Functions & Graphs - Module 4 and 5
- 1. Absolute Value Functions & Graphs Parent function of Abs. Functions y = |x|
Editor's Notes
- This is section 2-5 in the text.