This document contains a practice test with multiple choice and written response questions about transformations of graphs of functions. Some key questions ask students to:
1) Identify which statement about transforming a graph is false.
2) Determine if reflecting a function's graph in the y-axis will produce its inverse.
3) Write equations of functions after transformations like translations, stretches, and reflections.
4) Sketch and describe transformations of a graph to satisfy a given equation.
5) Determine the domain and range of a transformed function.
6) Find the inverse of a function and restrict its domain to make the inverse a function.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
This document contains lecture notes on inverse trigonometric functions. It begins with definitions of inverse functions and conditions for a function to have an inverse. It then defines the inverse trigonometric functions arcsin, arccos, arctan, and arcsec and gives their domains and ranges. Examples are provided to illustrate calculating values of the inverse trigonometric functions. The document concludes with a brief discussion of notational ambiguity and an outline mentioning derivatives of inverse trigonometric functions and applications.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
The document discusses drawing graphs of different types of functions, including linear, quadratic, cubic, and reciprocal functions. It provides the general forms of each type of function, describes the steps to draw their graphs which include constructing a table of values, plotting points, and joining the points. As an example, it shows the graph of a reciprocal function f(x) = 1/x for -1 ≤ x ≤ 1.5, which forms a hyperbola shape.
The document contains 20 multiple choice questions from an exam for the Naval School in 2017. The questions cover topics such as combinations, functions, limits, integrals, complex numbers, and geometry.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
This document provides notes and formulae on additional mathematics for Form 5. It covers topics such as progressions, integration, vectors, trigonometric functions, and probability. For progressions, it defines arithmetic and geometric progressions and gives the formulas for calculating the nth term and sum of terms. For integration, it provides rules and formulas for integrating polynomials, trigonometric functions, and expressions with ax+b. It also defines vectors and their operations including vector addition and subtraction. Other sections cover trigonometric functions, their definitions, relationships and graphs, as well as probability topics such as calculating probabilities of events and distributions like the binomial.
This document contains a review of graph transformations including translations, reflections, stretches, and compressions. It provides examples of transforming the graphs of functions by sketching the original graph and the images resulting from various combinations of transformations. It also includes determining equations to describe the transformed graphs.
This document contains lecture notes on inverse trigonometric functions. It begins with definitions of inverse functions and conditions for a function to have an inverse. It then defines the inverse trigonometric functions arcsin, arccos, arctan, and arcsec and gives their domains and ranges. Examples are provided to illustrate calculating values of the inverse trigonometric functions. The document concludes with a brief discussion of notational ambiguity and an outline mentioning derivatives of inverse trigonometric functions and applications.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
The document discusses drawing graphs of different types of functions, including linear, quadratic, cubic, and reciprocal functions. It provides the general forms of each type of function, describes the steps to draw their graphs which include constructing a table of values, plotting points, and joining the points. As an example, it shows the graph of a reciprocal function f(x) = 1/x for -1 ≤ x ≤ 1.5, which forms a hyperbola shape.
The document contains 20 multiple choice questions from an exam for the Naval School in 2017. The questions cover topics such as combinations, functions, limits, integrals, complex numbers, and geometry.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
This document provides notes and formulae on additional mathematics for Form 5. It covers topics such as progressions, integration, vectors, trigonometric functions, and probability. For progressions, it defines arithmetic and geometric progressions and gives the formulas for calculating the nth term and sum of terms. For integration, it provides rules and formulas for integrating polynomials, trigonometric functions, and expressions with ax+b. It also defines vectors and their operations including vector addition and subtraction. Other sections cover trigonometric functions, their definitions, relationships and graphs, as well as probability topics such as calculating probabilities of events and distributions like the binomial.
The document is a mathematics exam for Form 3 students on the topic of graphs of functions. It contains 35 multiple choice questions testing students' understanding of key concepts like linear functions, independent and dependent variables, interpreting graphs of functions, and the relationship between algebraic equations and their graphs. It also contains 5 short answer questions asking students to explain terms and describe graphs.
The document discusses different types of transformations of graphs: translations, reflections, and one-way stretches. It provides examples and explanations of how translations move a graph horizontally or vertically, reflections flip the graph across an axis, and stretches change the scale of the graph in the x or y direction. Key transformations include: y = f(x) + b for vertical translation; y = f(x - a) for horizontal translation; y = -f(x) for reflection across the x-axis; and y = df(x) for a stretch by a scale factor d in the y direction.
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.
This document appears to be an assignment containing multiple choice and short answer questions relating to calculus and 3D surfaces. It includes 20 questions testing the student's ability to: 1) identify equations of 3D surfaces from descriptions or sketches; 2) interpret formulas for drug concentration and car rental costs; and 3) match graphs or diagrams to functions. The assignment is due on October 19, 2014 at 11:47pm CDT.
This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.
This document contains practice problems and solutions for combining functions. It includes:
1. Multiple choice questions about compositions of functions.
2. Explicit equations for compositions and composite functions using given functions f(x), g(x), h(x), and k(x).
3. Graphing composite functions and determining their domains.
4. Evaluating composite functions for given values of x.
5. Writing composite functions as sums or compositions of simpler functions.
The document describes various transformations that can be applied to a graph y=f(x) to generate other graphs. These transformations include shifting the graph up or down by adding or subtracting a constant a to y (vertical shift), shifting the graph left or right by adding or subtracting a constant a to x (horizontal shift), reflecting the graph across the x-axis or y-axis, reflecting only parts of the graph where x or y is positive or negative, and stretching the graph vertically by multiplying y by a constant k. Examples of each transformation are shown through modified graphs.
The document is about quadratic polynomial functions and contains the following information:
1. It discusses investigating relationships between numbers expressed in tables to represent them in the Cartesian plane, identifying patterns and creating conjectures to generalize and algebraically express this generalization, recognizing when this representation is a quadratic polynomial function of the type y = ax^2.
2. It provides examples of converting algebraic representations of quadratic polynomial functions into geometric representations in the Cartesian plane, distinguishing cases in which one variable is directly proportional to the square of the other, using or not using software or dynamic algebra and geometry applications.
3. It discusses characterizing the coefficients of quadratic functions, constructing their graphs in the Cartesian plane, and determining their zeros (
This document contains 6 math problems involving graphing functions. Each problem has parts that involve:
1) Completing a table of values for a function.
2) Graphing the function on graph paper using given scales.
3) Finding specific values from the graph.
4) Drawing and finding values from a linear function related to the original.
The problems provide practice graphing and extracting information from graphs of quadratic, cubic, rational, and other polynomial functions. The document demonstrates how to set up and solve multi-step math word problems involving graphing functions.
The document discusses transformations of quadratic functions, including horizontal and vertical translations, reflections, and stretches or compressions. Horizontal translations move the graph right or left, depending on the value of h in the function f(x) = (x - h)2. Vertical translations move the graph up or down depending on the value of k. The vertex of the parabola after any transformation is located at the point (h, k). Reflections occur when the value of a in the function f(x) = a(x)2 is negative, causing the graph to reflect over the x-axis. Stretches and compressions occur when the absolute value of a is greater or less than 1, respectively.
The document defines definite integrals and discusses their properties. It states that a definite integral evaluates to a single number by integrating a function over a closed interval from a lower limit to an upper limit. It gives examples of using definite integrals to find areas bounded by curves. The mean value theorem for integrals is also introduced, which states that there is a rectangle between the inscribed and circumscribed rectangles with an area equal to the region under the curve. Exercises are provided on evaluating definite integrals and applying the mean value theorem.
This document discusses Karnaugh maps and their use in simplifying Boolean functions. It begins with an introduction to Karnaugh maps, including how they are constructed for 2, 3, and 4 variable functions. It then discusses how to use Karnaugh maps to find the simplest sum-of-products expression for a Boolean function by grouping adjacent 1's in the map. Examples are provided to demonstrate algebraic simplification and simplification using Karnaugh maps.
Karnaugh maps are a graphical technique used to simplify Boolean logic equations. They represent truth tables in a two-dimensional layout where physically adjacent cells imply logical adjacency. This adjacency allows common terms to be factored out to minimize logic expressions. Karnaugh maps are most commonly used to manually minimize logic with up to four variables into sum-of-products or product-of-sums form.
This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.
The document discusses key features to notice when analyzing graphs of functions. It identifies basic curve shapes defined by common equations, such as straight lines, parabolas, cubics, and circles. It also covers concepts like odd and even functions, symmetry, and dominance - how certain terms in an equation become more prominent at higher values of x. Basic curve shapes and these analytical concepts can help recognize and sketch the shape of graphs defined by functions.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Radical functions are similar to parabolas in that they can be transformed through vertical and horizontal shifts as well as reflections across the x-axis and y-axis. An example radical function is graphed as y = √x - 2 to demonstrate a vertical shift of -2 units. Another example function, y = 3√-(x - 1), contains a horizontal shift of 1 unit to the left, a vertical stretch by a factor of 3, and a reflection across the y-axis.
This document provides examples of using the fundamental counting principle to calculate the number of possible outcomes in various situations:
1) It shows how to calculate the number of possible outfit combinations based on the number of shirt and short options.
2) It demonstrates calculating the number of upholstery-color choices based on the number of upholstery and color options.
3) Other examples include calculating the number of possible computer systems, meal combinations, codes for a garage door opener, and postal codes.
4) The document emphasizes that the fundamental counting principle applies when tasks are related by "AND" but not when related by "OR", in which case probabilities must be added.
The document is a mathematics exam for Form 3 students on the topic of graphs of functions. It contains 35 multiple choice questions testing students' understanding of key concepts like linear functions, independent and dependent variables, interpreting graphs of functions, and the relationship between algebraic equations and their graphs. It also contains 5 short answer questions asking students to explain terms and describe graphs.
The document discusses different types of transformations of graphs: translations, reflections, and one-way stretches. It provides examples and explanations of how translations move a graph horizontally or vertically, reflections flip the graph across an axis, and stretches change the scale of the graph in the x or y direction. Key transformations include: y = f(x) + b for vertical translation; y = f(x - a) for horizontal translation; y = -f(x) for reflection across the x-axis; and y = df(x) for a stretch by a scale factor d in the y direction.
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.
This document appears to be an assignment containing multiple choice and short answer questions relating to calculus and 3D surfaces. It includes 20 questions testing the student's ability to: 1) identify equations of 3D surfaces from descriptions or sketches; 2) interpret formulas for drug concentration and car rental costs; and 3) match graphs or diagrams to functions. The assignment is due on October 19, 2014 at 11:47pm CDT.
This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.
This document contains practice problems and solutions for combining functions. It includes:
1. Multiple choice questions about compositions of functions.
2. Explicit equations for compositions and composite functions using given functions f(x), g(x), h(x), and k(x).
3. Graphing composite functions and determining their domains.
4. Evaluating composite functions for given values of x.
5. Writing composite functions as sums or compositions of simpler functions.
The document describes various transformations that can be applied to a graph y=f(x) to generate other graphs. These transformations include shifting the graph up or down by adding or subtracting a constant a to y (vertical shift), shifting the graph left or right by adding or subtracting a constant a to x (horizontal shift), reflecting the graph across the x-axis or y-axis, reflecting only parts of the graph where x or y is positive or negative, and stretching the graph vertically by multiplying y by a constant k. Examples of each transformation are shown through modified graphs.
The document is about quadratic polynomial functions and contains the following information:
1. It discusses investigating relationships between numbers expressed in tables to represent them in the Cartesian plane, identifying patterns and creating conjectures to generalize and algebraically express this generalization, recognizing when this representation is a quadratic polynomial function of the type y = ax^2.
2. It provides examples of converting algebraic representations of quadratic polynomial functions into geometric representations in the Cartesian plane, distinguishing cases in which one variable is directly proportional to the square of the other, using or not using software or dynamic algebra and geometry applications.
3. It discusses characterizing the coefficients of quadratic functions, constructing their graphs in the Cartesian plane, and determining their zeros (
This document contains 6 math problems involving graphing functions. Each problem has parts that involve:
1) Completing a table of values for a function.
2) Graphing the function on graph paper using given scales.
3) Finding specific values from the graph.
4) Drawing and finding values from a linear function related to the original.
The problems provide practice graphing and extracting information from graphs of quadratic, cubic, rational, and other polynomial functions. The document demonstrates how to set up and solve multi-step math word problems involving graphing functions.
The document discusses transformations of quadratic functions, including horizontal and vertical translations, reflections, and stretches or compressions. Horizontal translations move the graph right or left, depending on the value of h in the function f(x) = (x - h)2. Vertical translations move the graph up or down depending on the value of k. The vertex of the parabola after any transformation is located at the point (h, k). Reflections occur when the value of a in the function f(x) = a(x)2 is negative, causing the graph to reflect over the x-axis. Stretches and compressions occur when the absolute value of a is greater or less than 1, respectively.
The document defines definite integrals and discusses their properties. It states that a definite integral evaluates to a single number by integrating a function over a closed interval from a lower limit to an upper limit. It gives examples of using definite integrals to find areas bounded by curves. The mean value theorem for integrals is also introduced, which states that there is a rectangle between the inscribed and circumscribed rectangles with an area equal to the region under the curve. Exercises are provided on evaluating definite integrals and applying the mean value theorem.
This document discusses Karnaugh maps and their use in simplifying Boolean functions. It begins with an introduction to Karnaugh maps, including how they are constructed for 2, 3, and 4 variable functions. It then discusses how to use Karnaugh maps to find the simplest sum-of-products expression for a Boolean function by grouping adjacent 1's in the map. Examples are provided to demonstrate algebraic simplification and simplification using Karnaugh maps.
Karnaugh maps are a graphical technique used to simplify Boolean logic equations. They represent truth tables in a two-dimensional layout where physically adjacent cells imply logical adjacency. This adjacency allows common terms to be factored out to minimize logic expressions. Karnaugh maps are most commonly used to manually minimize logic with up to four variables into sum-of-products or product-of-sums form.
This document discusses functions and their properties. It defines a function as a special relation where each first element is paired with exactly one second element. Functions are represented as sets of ordered pairs. The domain of a function is the set of all possible x-values, while the range is the set of all possible y-values. Functions can be represented graphically and through equations, and can be transformed through shifts, reflections, and stretching/shrinking. Common function families include linear, quadratic, exponential, and trigonometric functions.
The document discusses key features to notice when analyzing graphs of functions. It identifies basic curve shapes defined by common equations, such as straight lines, parabolas, cubics, and circles. It also covers concepts like odd and even functions, symmetry, and dominance - how certain terms in an equation become more prominent at higher values of x. Basic curve shapes and these analytical concepts can help recognize and sketch the shape of graphs defined by functions.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
Radical functions are similar to parabolas in that they can be transformed through vertical and horizontal shifts as well as reflections across the x-axis and y-axis. An example radical function is graphed as y = √x - 2 to demonstrate a vertical shift of -2 units. Another example function, y = 3√-(x - 1), contains a horizontal shift of 1 unit to the left, a vertical stretch by a factor of 3, and a reflection across the y-axis.
This document provides examples of using the fundamental counting principle to calculate the number of possible outcomes in various situations:
1) It shows how to calculate the number of possible outfit combinations based on the number of shirt and short options.
2) It demonstrates calculating the number of upholstery-color choices based on the number of upholstery and color options.
3) Other examples include calculating the number of possible computer systems, meal combinations, codes for a garage door opener, and postal codes.
4) The document emphasizes that the fundamental counting principle applies when tasks are related by "AND" but not when related by "OR", in which case probabilities must be added.
The document is dated February 11, 2013 and appears to be a continuation of section 1.3. It discusses a meeting held on that date regarding an unnamed project and lists five action items that were agreed upon.
The document discusses the radian measure for angles. It explains that a radian is the central angle that intercepts an arc equal in length to the radius of the circle. A full circle contains 2π radians. Common angles like 90, 45, and 30 degrees can be expressed in radians as π/2, π/4, and π/6 respectively. Various formulas involving radians are provided, such as the circumference of a circle equals 2πr.
This document contains corrections for an inequalities test. It provides answers and explanations for problems on a short answer test regarding inequalities. The corrections cover multiple problems and explanations related to inequalities.
This document discusses inequalities involving quadratic and linear expressions in two variables. It provides examples of writing inequalities to describe graphs and relationships between two numbers. The document instructs students to complete homework problems involving solving and graphing various inequalities, and notifies them of a quiz the next day.
This document provides details about a meeting that took place on February 15, 2013. It lists the attendees which included John, Sarah, Mark and Lisa. The main topics discussed were the status of three ongoing projects and the schedule and budget for a new project.
This document provides examples and exercises for combining functions algebraically. It gives examples of combining two functions using addition, subtraction, multiplication, division, and composition. For each combination, it provides the explicit equation and determines the domain and range. It asks the reader to write explicit equations and determine domains and ranges for various combinations of functions, such as f(x) = x2 - 4 and g(x) = x - 1. The document also explores how the commutative, associative, and distributive properties apply when combining functions algebraically.
This document contains 3 entries from April 12, 2013. The first entry is dated Apr 12-2:20 PM. The second entry is dated Apr 12-2:39 PM. The third and final entry is dated Apr 12-2:54 PM.
This document discusses financial mathematics concepts like simple and compound interest. It provides examples of calculating future values of investments using the formula A = P(1+rt) for different principal amounts, interest rates, and time periods. For example, an investment of $1000 at 5% interest for 5 years will grow to a future value of $1250. The document also shows how to calculate rates of return and solves other related problems.
This document contains solutions to exercises about translating graphs of functions. It includes:
1) Examples of translating the graph of y = |x| by different amounts, and writing the equations of the translated graphs.
2) Describing how graphs are translated based on the equations of the form y = f(x - h) or y - k = f(x).
3) Sketching translated graphs on grids and stating their domains and ranges.
4) An example of finding the coordinates of the image of a point on a graph after it is translated.
5) Discussion of how vertical and horizontal asymptotes are affected by translation, and writing the equations of asymptotes for a
This document contains solutions to checkpoint questions about transforming graphs of functions. It includes examples of translating graphs by shifting them horizontally and vertically based on changes to the x and y variables in the function. It also contains an example of reflecting a graph across the x-axis. The questions require sketching the transformed graphs on grids and writing the equations of the transformed functions based on the given transformations.
1. The graph scale-change theorem states that multiplying the x- and y-values of a graph by constants n and m respectively will result in the same graph as dividing the equation variables x and y by n and m.
2. Applying a horizontal scale factor changes the width of the graph, while applying a vertical scale factor changes the height. Applying the same scale factor to both results in an overall size change.
3. Reflecting a graph over an axis by using a negative scale factor reverses the direction of the graph along that axis.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
1. This document discusses linear equations, slope, graphing lines, writing equations in slope-intercept form, and solving systems of linear equations.
2. Key concepts explained include slope as rise over run, the different forms of writing a linear equation, finding the x- and y-intercepts, and using two points to write the equation of a line in slope-intercept form.
3. Examples are provided to demonstrate how to graph lines based on their equations in different forms, find intercepts, write equations from two points, and solve systems of linear equations.
The document discusses graph translations. It defines a translation as moving each point (x,y) on a graph to a new location (x+h, y+k) using horizontal and vertical magnitudes h and k. Examples show finding the rule for a translation based on a point mapping, translating points between graphs, and rewriting equations in vertex form to identify translations between parabolas. The key aspects are that translations shift graphs in the xy-plane using h and k values and the graph-translation theorem allows identifying translations by replacing x with x-h and y with y-k in equations.
The document provides instructions on graphing parabolas using vertex form and translations. It defines the vertex form of a parabola as y = a(x - h)2 + k, where (h, k) are the coordinates of the vertex. Examples show how to find the image of a parabola under a translation Th,k and graph parabolas by hand by determining the vertex and symmetrical y-values. Steps are given to graph a parabola as finding the vertex, symmetrical values, and filling in the graph.
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
The document discusses various graphing techniques including stretching and shrinking graphs vertically or horizontally, reflecting graphs across axes, translating graphs vertically or horizontally, and identifying even, odd, and neither types of functions. It provides examples of how to determine if a graph is symmetric with respect to axes or the origin. Combinations of transformations are also discussed.
This document contains solutions to exercises from a pre-calculus textbook on radical functions.
1) It provides tables, graphs and explanations for various radical functions such as √x, √x+3, and their relation to other functions.
2) Students are asked to sketch graphs of radical functions based on given quadratic, cubic or other functions, and identify domains and ranges.
3) Radical equations are solved by graphing related functions and finding the x-intercept(s).
This document provides solutions to review problems involving combining functions through addition, subtraction, multiplication, division, and composition. Some key examples include:
- Sketching the graphs of f(x) + g(x), f(x) - g(x), f(x) * g(x), and f(x) / g(x) given the graphs of f(x) and g(x)
- Writing explicit equations for combinations of functions and determining their domains and ranges
- Evaluating composite functions like f(g(x)) and g(f(x)) given definitions of f(x) and g(x)
- Determining if two functions are inverses using their compositions
1. The document discusses various properties of graphs including symmetry, even and odd functions, translations, reflections, and stretching or compressing graphs.
2. It provides examples of applying these properties to determine if a graph is symmetric, even or odd, or to sketch graphs based on translations, reflections, or vertical/horizontal stretching and compressing of a original graph.
3. Several exercises are provided to apply these graph properties and transformations to specific functions and points on graphs.
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps such as graphing the individual functions separately first before combining using ordinates, examining the sign of products, and identifying asymptotes. An example for each operation is worked through step-by-step.
This document contains a math lab exercise on graphing rational functions with 4 questions. Question 1 has students predict holes in graphs of rational functions. Question 2 has students predict vertical asymptotes. Question 3 has students predict which graphs have horizontal asymptotes. The document provides step-by-step solutions and explanations for each question.
The document provides instructions and questions for a 2 hour math exam on graphs of functions. It includes 6 questions, each with parts a) to complete a table of values, b) to graph the function, c) to find specific values from the graph, and d) to find x-values from a related equation. The answers provide the completed tables, specific values found from the graphs, and x-values satisfying the related equations.
The document contains 10 math problems involving graphing functions and inequalities on Cartesian planes. The problems involve sketching graphs of functions, finding coordinates that satisfy equations, drawing lines to solve equations, and shading regions defined by inequalities. Tables are used to list x and y values satisfying equations.
The document outlines methods for graphing functions that involve addition, subtraction, multiplication, and division of other functions. It provides steps such as graphing the individual functions separately first before combining them based on the operation. Examples are given to illustrate each method, including identifying points where the individual functions are equal to 0 or 1 and investigating asymptotes.
There are four types of conic sections:
1. Ellipses have an equation of the form (x^2/a^2) + (y^2/b^2) = 1. The values of a and b determine the shape and size.
2. Hyperbolas have a similar equation to ellipses but with a minus sign instead of plus, resulting in two branches that extend to infinity.
3. Circles have the equation (x-a)^2 + (y-b)^2 = r^2, where (a,b) are the coordinates of the center and r is the radius.
4. Parabolas have the general form ax
This document provides examples and explanations for identifying key properties of quadratic functions presented in standard form, including:
1. Determining whether the graph opens upward or downward based on the sign of the leading coefficient a.
2. Finding the axis of symmetry by setting b/2a equal to x.
3. Finding the vertex by substituting the x-value from the axis of symmetry into the original function to determine the y-value.
4. Identifying the y-intercept from the constant term c.
Steps are demonstrated for graphing quadratic functions based on these properties, finding minimum/maximum values, and stating the domain and range. Examples analyze functions algebraically and graphically.
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
This document contains 3 short entries dated October 06, 2014 that are all labeled "6th october 2014". The document appears to be a log or record with multiple brief entries made on the same date.
This document is a list of dates, all occurring on October 3rd, 2014. Each entry repeats the date and contains a page number. There are 9 total entries in the list, each with the same date but incrementing page numbers from 1 through 9.
This document appears to be a log of dates from October 1st, 2014. It contains four entries all with the date October 1st, 2014 listed. The document provides a brief record of dates but does not include any other contextual information.
This document is a series of 8 entries all with the date of September 30, 2014. Each entry contains only the date with no other text or information provided.
The document is dated September 25, 2014. It appears to be a brief one paragraph document that does not provide much context or details. The date is the only substantive information given.
The document is dated September 25, 2014. It appears to be a brief one paragraph document that does not provide much context or details. The date is the only substantive information given.
This document is a record of events from September 24, 2014. It consists of 7 entries all with the same date of September 24, 2014 listed at the top, suggesting some type of daily log or journal was being kept for that date.
This document is a series of 7 entries all dated September 23, 2014 without any other notable information provided. Each entry simply states the date of September 23, 2014.
This four sentence document repeats the date September 22, 2014 four times without providing any additional context or information. The document states the same date, September 22, 2014, in each of its four sentences without elaborating on the significance of the date or including any other details.
This document is a record of dates, containing six identical entries of "September 18, 2014" with no other text or context provided. Each entry is on its own line and labeled with "18th sept 2014" and a number.
This document is a log of dates from September 16, 2014. It contains 5 entries all with the same date of September 16, 2014 listed in various formats including 16th sept 2014 and September 16, 2014.
This 3 sentence document simply repeats the date "September 11, 2014" three times on three different lines. It does not provide any other context or information.
The document is a list of dates, all occurring on September 9th, 2014. Each entry repeats the date 10 times, once for each numbered line. The sole purpose of the document is to repeatedly record the same date, September 9th, 2014, across 10 lines.
This document is a series of 7 entries all dated September 23, 2014 without any other notable information provided. Each entry simply states the date of September 23, 2014.
This four sentence document repeats the date September 22, 2014 four times without providing any additional context or information. The document states the same date, September 22, 2014, in each of its four sentences without elaborating on the significance of the date or including any other details.
This document is a record of dates, containing six identical entries of "September 18, 2014" with no other text or context provided. Each entry is on its own line and labeled with "18th sept 2014" and a number.
The document is a record of dates from September 17, 2014. It contains 20 entries, each listing the date September 17, 2014. The document functions as a log or record of the single date of September 17, 2014 recorded 20 separate times.
This document is a log of dates from September 16, 2014. It contains 5 entries all with the same date of September 16, 2014 listed in various formats including 16th sept 2014 and September 16, 2014.