This document contains the answer key for a math test on quadratic functions. It includes:
1) Graphing quadratic equations and finding vertices, zeros, and y-intercepts.
2) Solving quadratic equations by factoring.
3) Using the quadratic formula to solve equations.
4) Questions about salaries as a function of goals scored, the areas of trapezoids, and the maximum height of a tennis ball thrown in the air.
The document contains an answer key for a mathematics assignment on quadratic functions. It includes:
1) Graphing quadratic equations and identifying vertex, zeros, and y-intercept.
2) Solving quadratic equations by factoring.
3) Identifying true statements about quadratic functions.
4) Solving quadratic equations using the quadratic formula.
5) Setting up and solving an optimization word problem involving quadratic sales based on number of items sold.
6) Answering true/false questions based on a graph of a quadratic function.
7) Writing the quadratic equation for an age relationship problem.
8) Setting up the Pythagorean theorem to solve for side lengths of
This document contains an answer key for a quiz on quadratic functions. It includes:
1) Graphing quadratic equations and identifying vertex, zeros, axis of symmetry, and y-intercept.
2) Solving quadratic equations by factoring.
3) Identifying true statements about the discriminant and solutions of a quadratic equation.
4) Solving quadratic equations using the quadratic formula and identifying the discriminant and solutions.
5) Writing the equation for a situation involving profit from selling bracelets with discounts.
6) Writing the equation for a situation involving the sum of squares of consecutive even numbers.
7) Analyzing the graph of a quadratic equation to identify the vertex,
This document provides the final mark scheme for Edexcel's Core Mathematics C1 exam from January 2012. It lists the questions, schemes for awarding marks, and total marks for each question. The six mark questions cover topics like algebra, inequalities, coordinate geometry, and calculus. The longer questions involve multi-step problems applying these concepts, including sketching curves, finding equations of tangents and normals, and solving word problems involving formulas.
This document contains 14 math problems involving calculating the area under curves using definite integrals. The problems include finding the area under functions such as x2, √x, sec2x, and siny from given bounds. The areas are calculated and expressed as fractions or simplified numeric values.
1. This document contains examples of finding antiderivatives (indefinite integrals) of various functions.
2. The examples demonstrate using basic integration rules like power rule, reverse derivative rule, trigonometric integral formulas to find antiderivatives.
3. Additional examples show using initial conditions to determine an unknown constant term when an antiderivative is given in terms of an arbitrary constant.
This document discusses concepts and examples related to derivatives and critical points. It provides:
1) Examples of functions with critical points identified by setting the derivative equal to zero and finding the x-values that satisfy this. Maximum and minimum values are identified.
2) Practice problems for students to find the critical points, maximum/minimum values, and evaluate functions at critical points.
3) Additional examples illustrate situations where the derivative is undefined or does not equal zero at critical points.
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.
1. The document provides 13 mathematics questions covering topics such as: simplifying expressions, solving quadratic equations, working with cylinders and areas, and proving identities. Students must show their working and attempt all questions.
2. The questions involve skills like factorizing, using the quadratic formula, eliminating variables, finding minimum points on graphs, and proving statements about odd numbers. Working is required for full marks.
3. Students must complete the questions for homework and bring their work to the next mathematics lesson.
The document contains an answer key for a mathematics assignment on quadratic functions. It includes:
1) Graphing quadratic equations and identifying vertex, zeros, and y-intercept.
2) Solving quadratic equations by factoring.
3) Identifying true statements about quadratic functions.
4) Solving quadratic equations using the quadratic formula.
5) Setting up and solving an optimization word problem involving quadratic sales based on number of items sold.
6) Answering true/false questions based on a graph of a quadratic function.
7) Writing the quadratic equation for an age relationship problem.
8) Setting up the Pythagorean theorem to solve for side lengths of
This document contains an answer key for a quiz on quadratic functions. It includes:
1) Graphing quadratic equations and identifying vertex, zeros, axis of symmetry, and y-intercept.
2) Solving quadratic equations by factoring.
3) Identifying true statements about the discriminant and solutions of a quadratic equation.
4) Solving quadratic equations using the quadratic formula and identifying the discriminant and solutions.
5) Writing the equation for a situation involving profit from selling bracelets with discounts.
6) Writing the equation for a situation involving the sum of squares of consecutive even numbers.
7) Analyzing the graph of a quadratic equation to identify the vertex,
This document provides the final mark scheme for Edexcel's Core Mathematics C1 exam from January 2012. It lists the questions, schemes for awarding marks, and total marks for each question. The six mark questions cover topics like algebra, inequalities, coordinate geometry, and calculus. The longer questions involve multi-step problems applying these concepts, including sketching curves, finding equations of tangents and normals, and solving word problems involving formulas.
This document contains 14 math problems involving calculating the area under curves using definite integrals. The problems include finding the area under functions such as x2, √x, sec2x, and siny from given bounds. The areas are calculated and expressed as fractions or simplified numeric values.
1. This document contains examples of finding antiderivatives (indefinite integrals) of various functions.
2. The examples demonstrate using basic integration rules like power rule, reverse derivative rule, trigonometric integral formulas to find antiderivatives.
3. Additional examples show using initial conditions to determine an unknown constant term when an antiderivative is given in terms of an arbitrary constant.
This document discusses concepts and examples related to derivatives and critical points. It provides:
1) Examples of functions with critical points identified by setting the derivative equal to zero and finding the x-values that satisfy this. Maximum and minimum values are identified.
2) Practice problems for students to find the critical points, maximum/minimum values, and evaluate functions at critical points.
3) Additional examples illustrate situations where the derivative is undefined or does not equal zero at critical points.
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.
1. The document provides 13 mathematics questions covering topics such as: simplifying expressions, solving quadratic equations, working with cylinders and areas, and proving identities. Students must show their working and attempt all questions.
2. The questions involve skills like factorizing, using the quadratic formula, eliminating variables, finding minimum points on graphs, and proving statements about odd numbers. Working is required for full marks.
3. Students must complete the questions for homework and bring their work to the next mathematics lesson.
1. The document provides examples and explanations of key concepts in geometry including Cartesian coordinates, distance between points, types of triangles, area of triangles and polygons, division of line segments, slope and inclination of lines, and angle between two lines.
2. One example shows that the points (-2, 0), (2, 3) and (5, -1) are the vertices of a right triangle by applying the Pythagorean theorem.
3. Another example finds the area of the triangle with vertices (5, 4), (-2, 1) and (2, -3) to be 20 square units using the area formula.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
This document provides instructions for graphing linear equations. It begins with examples of solving linear equations algebraically. Students are then introduced to key properties of linear equations: they contain two variables and graph as straight lines. The document demonstrates graphing various linear equations by plotting their solution sets as points and connecting them with a straight line. It concludes by asking students to reflect on similarities and differences between the graphed linear equations.
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 provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
Exponential and logarithm functions are important in both theory and practice. They examine the graphs of exponential functions f(x)=ax where a>0 and logarithm functions f(x)=loga(x) where a>0. It is important to practice these functions so their properties become intuitive. Key properties include exponential functions where a>1 increase rapidly for positive x and 0<a<1 increase for decreasing negative x, and both pass through (0,1). The natural logarithm function f(x)=ln(x) is particularly important.
The document provides steps for graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x. Step 1 analyzes the monotonicity of the function by examining the sign chart of its derivative f'(x). Step 2 analyzes concavity by examining the sign chart of the second derivative f''(x). Step 3 combines these analyses into a single sign chart summarizing the function's properties over its domain. The goal is to completely graph the function, indicating zeros, asymptotes, critical points, maxima/minima, and inflection points.
1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs.
2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids.
3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.
This document provides instructions for graphing linear equations. It begins with a "Do Now" activity to find and correct mistakes in a sample function table and graph. It then shows examples of graphing different linear equations by plotting points from their function tables on coordinate grids. The document concludes with a prompt to explain to an absent student how to determine from a graph if a mistake was made in a problem.
1) The document is a teacher's notebook that provides instruction on the concept of slope. It includes examples of finding the slope between points on graphs, as well as graphs for the equations y=4x and y=-3x.
2) Students are asked to fill in tables of values and graph lines based on given equations in order to understand how changing the coefficient in the equation affects the slope of the line.
3) The key differences between the graphs of y=4x and y=-3x are explained - the equations have opposite signs for the coefficient, resulting in graphs with opposite slopes.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
1. The document discusses the concept of tangent lines and slope. It provides 5 examples of calculating the slope of a function at different points to derive the equation of the tangent line.
2. The slopes are calculated by taking the limit as h approaches 0 of the change in y over the change in x.
3. The slopes found were 2, 0, -1/2, 4, and 1/2, leading to tangent lines of y=2x-3, y=-2, y=-x/2+1, y=4x+2, and y=x/2+1 respectively.
This document provides instructions for a 45 minute MATLAB test with 6 questions worth a total of 25 marks. It instructs students to start MATLAB, load the test data, and provides some MATLAB commands that may be useful. It also notes that answers can be exact or numerical to 4 significant figures or decimal places.
5 marks scheme for add maths paper 2 trial spmzabidah awang
1. The document contains the mark scheme and solutions for an Additional Mathematics exam paper with 10 questions.
2. Question 1 involves solving a pair of simultaneous equations to find the values of x and y. Question 2 involves finding the maximum value and graph of a quadratic function.
3. Question 3 examines areas of shapes in a geometric progression. The final area is found to be 17066 cm^2.
4. Subsequent questions cover topics such as function graphs, probability distributions, logarithmic graphs, trigonometry, and simultaneous equations.
5. The last few questions deal with topics like probability, normal distribution, and kinematic equations. Overall the document provides a comprehensive breakdown of the marking scheme
Sqlsat154 maintain your dbs with help from ola hallengrenAndy Galbraith
Ola Hallengren created a free database maintenance solution that automates common maintenance tasks like backups, integrity checks, and index optimization. The solution uses stored procedures that can be easily scheduled. It has options to configure maintenance settings and log output. Implementing the solution involves downloading a script, configuring settings, and scheduling the provided jobs.
The document is a notebook containing timestamps from December 14, 2011 documenting various activities between 10:44 AM and 11:41 AM, including notes taken between 11:09 AM and 11:09 AM, 11:13 AM and 11:14 AM, and 11:17 AM and 11:19 AM. It is divided into 4 pages documenting the ongoing activities throughout the day.
The document appears to be notes from a math class, containing several word problems and their solutions. It includes questions about the number of teachers needed at a school of 850 students, workers needed to complete a job in a shorter time, finding an original number based on operations, and splitting $150 among three people with specified differences in amounts. The notes provide steps to solve each math problem presented.
This document contains 11 problems involving quadratic functions and equations. The problems cover graphing quadratic functions, solving quadratic equations by factoring and using the quadratic formula, identifying properties of quadratic functions, modeling real-world word problems with quadratic equations, and applying the Pythagorean theorem and properties of parabolas.
The document appears to be notes from an untitled notebook on December 7, 2011. It consists of time-stamped entries documenting the user's activities that day, including browsing websites and taking notes, from 10:48 AM to 11:45 AM.
1. The document provides examples and explanations of key concepts in geometry including Cartesian coordinates, distance between points, types of triangles, area of triangles and polygons, division of line segments, slope and inclination of lines, and angle between two lines.
2. One example shows that the points (-2, 0), (2, 3) and (5, -1) are the vertices of a right triangle by applying the Pythagorean theorem.
3. Another example finds the area of the triangle with vertices (5, 4), (-2, 1) and (2, -3) to be 20 square units using the area formula.
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
This document provides instructions for graphing linear equations. It begins with examples of solving linear equations algebraically. Students are then introduced to key properties of linear equations: they contain two variables and graph as straight lines. The document demonstrates graphing various linear equations by plotting their solution sets as points and connecting them with a straight line. It concludes by asking students to reflect on similarities and differences between the graphed linear equations.
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 provides a marking scheme for an Additional Mathematics paper 2 trial examination from 2010. It consists of 7 questions, each with multiple parts. For each question, it lists the number of marks awarded for various steps in the solutions, such as setting up the correct formula, performing calculations accurately, obtaining the right solution, plotting points correctly, and using appropriate mathematical reasoning. The highest number of marks for a single question is 8 marks. The marking scheme evaluates multiple aspects of students' work and reasoning for 7 multi-step mathematics problems.
Exponential and logarithm functions are important in both theory and practice. They examine the graphs of exponential functions f(x)=ax where a>0 and logarithm functions f(x)=loga(x) where a>0. It is important to practice these functions so their properties become intuitive. Key properties include exponential functions where a>1 increase rapidly for positive x and 0<a<1 increase for decreasing negative x, and both pass through (0,1). The natural logarithm function f(x)=ln(x) is particularly important.
The document provides steps for graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x. Step 1 analyzes the monotonicity of the function by examining the sign chart of its derivative f'(x). Step 2 analyzes concavity by examining the sign chart of the second derivative f''(x). Step 3 combines these analyses into a single sign chart summarizing the function's properties over its domain. The goal is to completely graph the function, indicating zeros, asymptotes, critical points, maxima/minima, and inflection points.
1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs.
2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids.
3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.
This document provides instructions for graphing linear equations. It begins with a "Do Now" activity to find and correct mistakes in a sample function table and graph. It then shows examples of graphing different linear equations by plotting points from their function tables on coordinate grids. The document concludes with a prompt to explain to an absent student how to determine from a graph if a mistake was made in a problem.
1) The document is a teacher's notebook that provides instruction on the concept of slope. It includes examples of finding the slope between points on graphs, as well as graphs for the equations y=4x and y=-3x.
2) Students are asked to fill in tables of values and graph lines based on given equations in order to understand how changing the coefficient in the equation affects the slope of the line.
3) The key differences between the graphs of y=4x and y=-3x are explained - the equations have opposite signs for the coefficient, resulting in graphs with opposite slopes.
The document discusses equations of straight lines. It covers determining the gradient and equation of a straight line, as well as representing a line in various forms such as y=mx+c, Ax + By + C = 0, and identifying lines from two points. Key concepts covered include finding the gradient from two points, the relationship between perpendicular lines, and deriving the equation of a line given values for the gradient and a point.
1. The document discusses the concept of tangent lines and slope. It provides 5 examples of calculating the slope of a function at different points to derive the equation of the tangent line.
2. The slopes are calculated by taking the limit as h approaches 0 of the change in y over the change in x.
3. The slopes found were 2, 0, -1/2, 4, and 1/2, leading to tangent lines of y=2x-3, y=-2, y=-x/2+1, y=4x+2, and y=x/2+1 respectively.
This document provides instructions for a 45 minute MATLAB test with 6 questions worth a total of 25 marks. It instructs students to start MATLAB, load the test data, and provides some MATLAB commands that may be useful. It also notes that answers can be exact or numerical to 4 significant figures or decimal places.
5 marks scheme for add maths paper 2 trial spmzabidah awang
1. The document contains the mark scheme and solutions for an Additional Mathematics exam paper with 10 questions.
2. Question 1 involves solving a pair of simultaneous equations to find the values of x and y. Question 2 involves finding the maximum value and graph of a quadratic function.
3. Question 3 examines areas of shapes in a geometric progression. The final area is found to be 17066 cm^2.
4. Subsequent questions cover topics such as function graphs, probability distributions, logarithmic graphs, trigonometry, and simultaneous equations.
5. The last few questions deal with topics like probability, normal distribution, and kinematic equations. Overall the document provides a comprehensive breakdown of the marking scheme
Sqlsat154 maintain your dbs with help from ola hallengrenAndy Galbraith
Ola Hallengren created a free database maintenance solution that automates common maintenance tasks like backups, integrity checks, and index optimization. The solution uses stored procedures that can be easily scheduled. It has options to configure maintenance settings and log output. Implementing the solution involves downloading a script, configuring settings, and scheduling the provided jobs.
The document is a notebook containing timestamps from December 14, 2011 documenting various activities between 10:44 AM and 11:41 AM, including notes taken between 11:09 AM and 11:09 AM, 11:13 AM and 11:14 AM, and 11:17 AM and 11:19 AM. It is divided into 4 pages documenting the ongoing activities throughout the day.
The document appears to be notes from a math class, containing several word problems and their solutions. It includes questions about the number of teachers needed at a school of 850 students, workers needed to complete a job in a shorter time, finding an original number based on operations, and splitting $150 among three people with specified differences in amounts. The notes provide steps to solve each math problem presented.
This document contains 11 problems involving quadratic functions and equations. The problems cover graphing quadratic functions, solving quadratic equations by factoring and using the quadratic formula, identifying properties of quadratic functions, modeling real-world word problems with quadratic equations, and applying the Pythagorean theorem and properties of parabolas.
The document appears to be notes from an untitled notebook on December 7, 2011. It consists of time-stamped entries documenting the user's activities that day, including browsing websites and taking notes, from 10:48 AM to 11:45 AM.
1. The document contains the answer key to a math test on quadratic functions.
2. It includes graphing quadratic equations, solving by factoring, identifying properties, using the quadratic formula, modeling real world situations, and applying the Pythagorean theorem and properties of parabolas.
3. Several questions involve finding the vertex, x-intercepts, maximum/minimum values, and distance or drop measures for quadratic equations describing real world motions or sales situations.
This document contains:
1) A key for a quadratic functions test with graphing and solving problems.
2) The test asks students to graph and solve quadratic equations, identify true statements about discriminants and solutions, and solve word problems involving quadratic models.
3) The key provides the full worked out solutions and answers to all problems on the test.
The document contains 23 math problems involving equations, inequalities, geometry concepts like angles and lengths of lines, limits, and other algebraic expressions. The problems cover a wide range of math topics including functions, polynomials, systems of equations, trigonometry, and calculus.
The document discusses graphing quadratic functions of the form f(x) = ax^2 + bx + c. The key points are:
1) The graph of any quadratic function is a parabola.
2) To graph f(x) = a(x - h)^2 + k, find the vertex (h, k), x-intercepts by setting f(x) = 0, y-intercept, and plot points to form the parabola shape.
3) The vertex of f(x) = ax^2 + bx + c is (-b/2a, f(-b/2a)), which is the minimum if a > 0 and maximum if a
The document contains a midterm exam for an ODE class with 6 problems worth 10 points each. Problem 1 asks to find the general solution of a 7th order linear ODE using the method of undetermined coefficients. Problem 2 asks to solve a 2nd order linear ODE using either variation of parameters or undetermined coefficients. Problem 3 asks to solve a nonlinear 2nd order ODE using a substitution. Problem 4 asks to find the equation of motion for a mass attached to a spring with an external force applied. Problem 5 asks to solve an eigenvalue problem for a CE equation. Problem 6 asks to use variation of parameters to solve a 2nd order nonhomogeneous ODE.
This document provides 24 two-step equations to solve. The solutions are provided in curly brackets next to each equation. The equations involve addition, subtraction, multiplication, division, and operations with variables.
The document discusses using functions to find output values from input values. It provides examples of functions in the form of y=fx(x) and has students complete function tables and plot points for various functions. It discusses how the Rube Goldberg cartoon from the beginning uses an input, output, and rule to demonstrate a function.
This document announces the release of Version 5 educational software containing over 15,000 presentation slides, 1,000 example/student questions, 100 worksheets, 1,200 interactive exercises, and 5,000 mental math questions across two CDs. It provides a 7-minute demo of 20 sample slides and directs users to register for a free account to access additional full presentations.
This document provides 20 algebra problems involving simplifying expressions, solving equations, graphing lines, writing equations in slope-intercept form, determining if lines are parallel or perpendicular, and operations with scientific notation. Students are instructed to show all work.
1. The document discusses functions of several variables and partial differentiation.
2. It provides examples of functions with different domains and ranges, such as f(x,y) = x^2 + y^2 which has a domain of all real numbers and a range of non-negative real numbers.
3. It also examines how changing variables in functions impacts the output, like how increasing humidity by 20% at 80 degrees Fahrenheit increases the heat index by about 3 units.
The document contains examples of functions of several variables and their domains and ranges. It provides equations for various functions and graphs their surfaces over different domains. Some key examples include functions defined by equations like x2 + y2 = 1, 2, 3 and functions where increasing one variable by a fixed amount increases the output by a fixed amount.
1) Simultaneous equations involve two variables in two equations that are solved simultaneously to find the values of the variables.
2) To solve simultaneous equations, one first expresses one variable in terms of the other by changing the subject of one linear equation, then substitutes this into the other equation to obtain a quadratic equation.
3) This quadratic equation is then solved using factorisation or the quadratic formula to find the values of the variables that satisfy both original equations.
This document provides an ASSURE model for a math lesson on graphing and interpolating quadratic equations for a high school Algebra 2 class. The lesson objectives are for students to correctly substitute points on a graph into a quadratic function 80% of the time and graph quadratic equations on a graphing calculator choosing an appropriate window 80% of the time. The lesson involves representing quadratic equations as graphs and equations, identifying key features of parabolas, using three points to find the equation of a graph, and practicing with different point sets.
1. This document contains notes and formulas for additional mathematics form 4. It covers topics such as quadratic equations, functions, indices and logarithms, coordinate geometry, and statistics.
2. Quadratic equations are discussed, including finding the roots of a quadratic equation and writing the equation from its roots. Quadratic functions are also covered, specifically the relationship between the sign of b^2 - 4ac and the nature of the roots.
3. Other topics include indices and logarithm laws, coordinate geometry concepts like distance and midpoints, statistics topics such as measures of central tendency (mean, median, mode), and measures of dispersion like standard deviation and interquartile range.
The document provides information and examples on linear laws and linear relationships. It discusses:
- Drawing lines of best fit by inspection of data points.
- Writing equations for lines of best fit in the form of y = mx + c.
- Determining values of variables from lines of best fit and equations.
- Reducing non-linear relationships to linear form by rearranging variables.
- Finding values of constants in non-linear relationships by plotting graphs of best fit lines and determining the gradient and y-intercept.
Worked examples are provided to illustrate key concepts like identifying dependent and independent variables, determining the gradient and y-intercept, and using these to solve for constants in non-
Calculo y geometria analitica (larson hostetler-edwards) 8th ed - solutions m...ELMIR IVAN OZUNA LOPEZ
This document contains the solutions to odd-numbered exercises from Chapter P of a calculus textbook. It provides answers and work for 43 problems involving graphing functions, finding intercepts, determining symmetries, and other skills related to functions and their graphs. The problems progress from simple linear functions to more complex expressions involving square roots, cubes, and other operations.
1. The document contains examples solving systems of linear equations and linear inequalities arising from word problems about mixtures, costs, graphs of lines, and similar contexts.
2. Similar figures and corresponding parts of congruent triangles are used to solve for missing lengths and angle measures.
3. Place value and binary and hexadecimal number systems are explained.
1) The volume of the solid region bounded by z = 9 - x^2 - y in the first octant is found using iterated integration.
2) The volume of the region bounded by z = x^2 + y^2, x^2 + y^2 = 25, and the xy-plane is found using polar coordinates.
3) The double integral of sin(x^2) over the region from 0 to 9 in x and y from 0 to x is evaluated.
1. Basic algebra involves variables, algebraic expressions, and equations. Variables represent unknown values.
2. Algebraic expressions contain variables, numbers, and operators. They can be simplified by combining like terms or using properties of exponents.
3. Equations set two algebraic expressions equal to each other and can be solved algebraically to find the value of variables. There are methods for solving different types of equations like linear, fractional, and simultaneous equations.
1. This document contains 11 multi-part math problems involving systems of equations and inequalities. The problems cover topics such as solving systems graphically, algebraically, and determining if ordered pairs are solutions. They also involve word problems about ages, expenses, and splitting amounts into parts.
2. Key steps addressed include setting up tables of values, identifying line types, finding the solution set intersection, using substitution or elimination methods, stating yes or no for ordered pairs, and drawing graphs of solution sets for systems of inequalities.
3. The problems progress from simpler systems to more complex ones involving multiple equations or inequalities, requiring skills like algebraic manipulation, graphical analysis, and translating word problems into mathematical systems.
1. The solution to the system of equations y=2x and x/5 is (0,0).
2. The solutions to the systems of equations x+y=5 and 3x+2y-14=0 are (3,2) and the solutions to x=3 and y=6.5 are (3,6.5).
3. The system of equations representing spending $164 on books costing $15 each or $17 each can be expressed as a system of first degree equations in two variables.
1) The document discusses solving systems of equations and inequalities. It contains 13 problems involving setting up, graphing and solving systems of linear and nonlinear equations and inequalities.
2) The problems cover a range of techniques for solving systems, including substitution, elimination, graphing, and applying constraints to identify variable values that satisfy simultaneous relationships.
3) The document provides practice with setting up and solving different types of systems, as well as interpreting solutions in the context of word problems about rates, prices, ages and coin values.
The document contains 14 problems involving systems of equations and inequalities. Problem 1 asks for the solution of a system of equations as an ordered pair. Problem 2 asks to determine the solution set of another system of equations. Problem 3 asks to express a word problem as a system of two equations.
This document contains 11 problems involving quadratic functions and equations. The problems cover graphing quadratic functions, solving quadratic equations by factoring and using the quadratic formula, identifying properties of quadratic functions, modeling real-world word problems with quadratic equations, and applying the Pythagorean theorem and properties of parabolas.
This document provides 10 problems involving quadratic functions and equations: (1) graphing quadratic equations; (2) solving quadratic equations by factoring; (3) identifying true statements about quadratic equations; (4) solving quadratic equations using the quadratic formula; (5) writing an equation to model profit from bracelet sales; (6) writing an equation for the sum of squares of two consecutive even numbers; (7) analyzing a graph of a quadratic function; (8) solving for measurements of a rhombus given information about its diagonals and area; (9) solving for the number of students in a classroom given information about class time and changes in time per student; (10) solving application problems involving the height
This document contains 10 problems related to quadratic functions and equations. The problems cover topics such as: 1) graphing quadratic functions, 2) solving quadratic equations by factoring and using the quadratic formula, 3) identifying properties of quadratic functions from their graphs or equations, and 4) modeling real-world situations using quadratic equations. The goal is to demonstrate understanding of key concepts for quadratic functions and practice applying various techniques for solving quadratic equations.
This document contains 10 word problems involving quadratic functions. Each problem provides relationships between two or more numbers and asks the reader to determine the specific values of those numbers based on the given information. The problems cover a variety of quadratic equation scenarios including differences, sums, products, and consecutive numbers.
The document provides 10 examples of word problems involving quadratic equations. For each problem it defines the variables, sets up the quadratic equation, solves for the zeros, and states the answer.
This document contains an answer key for a math worksheet on quadratic functions. It includes:
1) Graphing quadratic equations and finding vertices, zeros, and y-intercepts.
2) Solving quadratic equations by factoring.
3) Identifying true statements about the discriminant and solutions of quadratic equations.
4) Solving quadratic equations using the quadratic formula.
5) Writing the equation for a situation where a hockey player's salary is a quadratic function of his goals.
6) Analyzing the graph of a quadratic equation.
7) Writing the equation for a situation where the sum and product of two numbers is given.
8) Setting up and
This document provides 10 problems involving quadratic functions and equations: (1) graphing quadratic equations; (2) solving quadratic equations by factoring; (3) identifying true statements about quadratic equations; (4) solving quadratic equations using the quadratic formula; (5) writing an equation to model profit from bracelet sales; (6) writing an equation for the sum of squares of two consecutive even numbers; (7) analyzing a graph of a quadratic function; (8) finding measurements of diagonals of a rhombus using its area; (9) determining the number of students in a classroom using time per student; (10) analyzing the height of a hot air balloon over time.
This document contains 10 problems related to quadratic functions and equations. The problems cover topics such as: 1) graphing quadratic functions, 2) solving quadratic equations by factoring and using the quadratic formula, 3) identifying properties of quadratic functions from their graphs or equations, and 4) modeling real-world situations using quadratic equations. The goal is to demonstrate understanding of key concepts for quadratic functions and practice applying various techniques for solving quadratic equations.
1. MTH-4108 A
Quadratic Functions
ANSWER KEY
1. Graph the following equations. Be sure to include the coordinates of at least 5
points, including the vertex, the zeros (if any) and y-intercept.
a) y = 0.4x2 – 3x −b −∆
,
y 2a 4 a
Vertex: 3 − 9
,
0.8 1.6
( 3.75,−5.62)
−b± ∆
2a
Zeros: 3 ± 9
x
0.8
{ 0,7.5}
y = 0.4(0) − 3(0)
y-intercept:
y=0
b) y = –x2 + 5x – 6.25
y
Vertex:
−b −∆
,
2a 4a
− 5 − ( 25 − 4( − 1)( − 6.25) )
,
−2 −4
( 2.5,0)
x
−b± ∆
2a
Zeros: − 5 ± 0
−2
{ 2.5,0}
y-intercept:
y = −1(0) + 5(0) − 6.25
y = −6.25
2. c) y = 2x2 + 4x + 3 Vertex:
y −b −∆
,
2a 4a
− 4 − (16 − 4( 2)( 3) )
,
4 8
( − 1,1)
−b± ∆
2a
x Zeros: − 4 ± − 8
4
{ ∅}
y-intercept:
y = 2(0) + 4(0) + 3
y =3
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2. Solve the following equations by factoring:
a) 25x2 – 1 = 0 ( 5x − 1) ⇒ 5 x = 1 ⇒ x = 5
1
( 5x + 1) ⇒ 5 x = −1 ⇒ x = − 1 5
2 x 2 + 8 x − 5 x − 20 = 0
b) 2x2 + 3x – 20 = 0 2 x( x + 4 ) − 5( x + 4)
( 2 x − 5) ⇒ 2 x = 5 ⇒ x = 5 2 /5
( x + 4 ) ⇒ x = −4
3. CIRCLE the true statement(s) below:
a) If ∆ = 0 for a quadratic equation, it means that there are 2 solutions with
one solution equal to 0.
b) If the discriminant is less than 0, it means that there are no real solutions
for this equation.
c) If an equation has two zeros which are equal to each other it means that ∆ = 0
d) If an equation has one negative solution, it is impossible that the other
solution is positive if ∆ > 0.
e) If the discriminant is equal to zero then the equation has no solution.
/5
3. 4. Solve the following equations using the quadratic formula. Clearly indicate the
value of ∆ and round your answers to the nearest thousandth when necessary.
a) 0.5x2 + 4x – 5 = 0 −b± ∆
2a
Zeros: − 4 ± (16 − 4( 0.5)( − 5) )
1
{ − 9.099,1.099}
b) -x2 – x – 11.25 = 0 −b± ∆
3
2a
Zeros: 1 ± (1 − 4(− 13 )( − 11.25) )
−2
3
{ ∅}
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5. A professional hockey player scores 30 goals in one season and earns $1 800 000,
for an average salary of $60 000 per goal. His contract states that for each
additional goal, his average salary will increase by $100. The table below
illustrates his salary as a function of how many goals he scores.
No. of goals Number of Average salary per goal Total salary
scored after 30 total goals
0 30 $60 000 $1 800 000
1 30 + 1 = 31 60 000 + (100× 1) = $60 100 $1 863 100
2 30 + 2 = 32 60 000 + (100× 2) = $60 200 $1 926 400
x 30 + x 60 000 + (100× x) (30 + x)(60000 + 100x)
Write the equation in the form ax2 + bx + c which illustrates this situation.
( 30 + x )( 60000 + 100 x )
1800000 + 3000 x + 60000 x + 100 x 2
y = 100 x 2 + 63000 x + 1800000
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4. 6. Answer the following questions using the graph below:
0.5x2 – x – 4 = y
y
x
Vertex:
a) What are the coordinates of the vertex?
− b − ∆ 1 − (1 − 4( 0.5)( − 4 ) )
, ⇒ , ⇒ (1,−4.5)
2a 4a 1 2
b) What is the equation of the axis of symmetry?
x=1
c) Is there a maximum or a minimum?
Minimum of -4.5
d) What are the zeros?
−b± ∆ 1± 9
Zeros: ⇒ ⇒ { − 2,4}
2a 1
e) What is the y-intercept?
y-intercept: y = 0.5(0) − (0) − 4 ⇒ y = −4
/5
7. Without calculating, write the equation in the form ax2 + bx + c which illustrates
the following situation:
The sum of two numbers is 100 and their product is 2 356.
Let x = 1st number
Let 100 – x = 2nd number
x(100 − x ) = 2356
100 x − x 2 = 2356
x 2 − 100 x + 2356 = 0
/5
5. 8. The small base of a trapezoid measures double the height. The large base
measures 3 metres more than the small base and the area of the trapezoid is equal
to 123.75 metres squared. What are the measurements of the small base, large
base and height of this trapezoid?
Let x = height
Let 2x = small base
Let 2x + 3 = large base
x[ 2 x + ( 2 x + 3) ]
= 123.75 −b± ∆
2
2a
x[ 4 x + 3]
= 123.75 − 3 ± 9 − 4( 4 )( − 247.5)
2
Zeros: 8
4 x 2 + 3 x 123.75
= − 3 ± 3969
2 1
4 x + 3 x = 247.5
2 8
{ − 8.25,7.5}
4 x 2 + 3 x − 247.5 = 0
/5
9. A sports store sells a certain number of bicycles at the regular price and receives
$17 500 profit. The following week the bikes go on sale for $150 less each
bicycle, and the store sells 15 more bicycles for the same total profit as the
previous week. What is the regular price of one bicycle?
Let x = price of one bicycle
17500 17500
+ 15 = −b± ∆
x x − 150
17500 + 15 x 17500 2a
=
x x − 150 150 ± 22500 − 4(1)( − 175000 )
17500 x − 2625000 + 15 x 2 − 2250 x = 17500 x Zeros: 2
150 ± 850
15 x 2 − 2250 x − 2625000 = 0
2
{ − 350,500}
x 2 − 150 x − 175000 = 0
/10
6. 10. We throw a tennis ball off the school roof. The equation of the height (y) of the
ball in metres is: y = 10 + 8.75t – 5t2, where t represents the time in minutes.
a) What is the maximum height obtained by the ball?
Vertex:
− b − ∆ − 8.75 − ( 76.5625 − 4( − 5)(10 ) )
, ⇒ , ⇒ ( 0.875,13.8)
2a 4a − 10 − 20
b) After how many minutes is the maximum height obtained?
Vertex:
− b − ∆ − 8.75 − ( 76.5625 − 4( − 5)(10 ) )
, ⇒ , ⇒ ( 0.875,13.8)
2a 4a − 10 − 20
c) If the school is 7m high, after how many minutes does the ball land?
7 = −5t 2 + 8.75t + 10
y-coordinate:
0 = −5t 2 + 8.75t + 3
−b± ∆ − 8.75 ± 8.752 − 4( − 5)( 3) − 8.75 ± 136.5625
⇒ ⇒
Zeros: 2a − 10 − 10
{ − 0.294,2.044}
Round your answers to the nearest hundredth.
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