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2.6
There are families of functions that share characteristics, with a parent function being the simplest form. Transformations of the parent function include translations, which shift the graph horizontally and/or vertically without changing shape; reflections, which flip the graph over an axis; and vertical stretches or compressions, which multiply or reduce the y-values by a factor. Combining transformations results in a new function, and the document provides examples of determining rules for functions related through transformations.
How to be Successful in Stats
The document provides tips for being successful in a statistics course. It recommends making a study plan, preparing for and attending class, taking notes, doing homework, finding a study buddy, keeping up with the work, preparing for tests and quizzes, and learning from mistakes. Key tips include attending class, practicing by doing homework, not getting behind, and reviewing past material in preparation for tests and quizzes.
2.1 Relations and Functions
This document discusses functions, equations, and graphs. It defines relations as sets of input-output pairs that can be represented in four ways. A function is a special type of relation where each input corresponds to exactly one output. The vertical line test can be used to determine if a graph represents a function - if any vertical line crosses more than one point, it is not a function. Function notation represents an output value in terms of an input value using an equation. Examples show how to write function rules, evaluate functions, and use the vertical line test to identify functions from graphs.
5.2
This document discusses polynomials, factors, zeros, and graphing polynomial functions. It contains the following key points:
1) A polynomial can be written in factored form by factoring out the greatest common factor and setting each linear factor equal to zero. The zeros of a polynomial function are its x-intercepts.
2) The Factor Theorem states that an expression x - a is a factor of a polynomial if and only if a is a zero of the related polynomial function.
3) To write a polynomial function given its zeros, write each zero as a linear factor and multiply the factors together. Multiple zeros have a linear factor that is repeated, called the multiplicity. The multiplicity indicates if the
7.3
This document discusses how to construct confidence intervals using point estimates, margins of error, and a calculator. It provides instructions for using the calculator to find confidence intervals from sample data, including entering the number of successes and trials and calculating the solution. Several assignment problems are listed for practice.
6.3
This document discusses working with binomial radical expressions, including:
- Combining like radicals using the distributive property by adding or subtracting the coefficients
- Simplifying expressions by reducing radicals first before combining like terms
- Multiplying binomials with radical expressions using FOIL and being careful when reducing and combining like terms
- Multiplying conjugates, where expressions only differ in the signs of the second terms
- Rationalizing denominators involving square roots by multiplying the numerator and denominator by the conjugate of the denominator
Ch 1 review stations CW answers
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
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2.6
There are families of functions that share characteristics, with a parent function being the simplest form. Transformations of the parent function include translations, which shift the graph horizontally and/or vertically without changing shape; reflections, which flip the graph over an axis; and vertical stretches or compressions, which multiply or reduce the y-values by a factor. Combining transformations results in a new function, and the document provides examples of determining rules for functions related through transformations.
How to be Successful in Stats
The document provides tips for being successful in a statistics course. It recommends making a study plan, preparing for and attending class, taking notes, doing homework, finding a study buddy, keeping up with the work, preparing for tests and quizzes, and learning from mistakes. Key tips include attending class, practicing by doing homework, not getting behind, and reviewing past material in preparation for tests and quizzes.
2.1 Relations and Functions
This document discusses functions, equations, and graphs. It defines relations as sets of input-output pairs that can be represented in four ways. A function is a special type of relation where each input corresponds to exactly one output. The vertical line test can be used to determine if a graph represents a function - if any vertical line crosses more than one point, it is not a function. Function notation represents an output value in terms of an input value using an equation. Examples show how to write function rules, evaluate functions, and use the vertical line test to identify functions from graphs.
5.2
This document discusses polynomials, factors, zeros, and graphing polynomial functions. It contains the following key points:
1) A polynomial can be written in factored form by factoring out the greatest common factor and setting each linear factor equal to zero. The zeros of a polynomial function are its x-intercepts.
2) The Factor Theorem states that an expression x - a is a factor of a polynomial if and only if a is a zero of the related polynomial function.
3) To write a polynomial function given its zeros, write each zero as a linear factor and multiply the factors together. Multiple zeros have a linear factor that is repeated, called the multiplicity. The multiplicity indicates if the
7.3
This document discusses how to construct confidence intervals using point estimates, margins of error, and a calculator. It provides instructions for using the calculator to find confidence intervals from sample data, including entering the number of successes and trials and calculating the solution. Several assignment problems are listed for practice.
6.3
This document discusses working with binomial radical expressions, including:
- Combining like radicals using the distributive property by adding or subtracting the coefficients
- Simplifying expressions by reducing radicals first before combining like terms
- Multiplying binomials with radical expressions using FOIL and being careful when reducing and combining like terms
- Multiplying conjugates, where expressions only differ in the signs of the second terms
- Rationalizing denominators involving square roots by multiplying the numerator and denominator by the conjugate of the denominator
Ch 1 review stations CW answers
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Ch 8 test review key
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help alleviate symptoms of mental illness and boost overall mental well-being.
6.1
This document discusses roots and radicals. It defines that for every power, there is a corresponding root. It explains that the nth root of a number a is the number that when raised to the nth power equals a. It discusses that if n is odd there is one real nth root, and if n is even and the number is positive there are two real nth roots, and if the number is negative there are no real nth roots. It provides examples of finding cube and fourth roots. It also discusses simplifying radicals and working with variables in radicals.
4.7 The Quadratic Formula
This document discusses solving quadratic equations using the quadratic formula. It begins by reviewing previous methods like factoring and graphing. It then introduces the quadratic formula, which can be used to solve any quadratic equation in standard form. Examples are provided to demonstrate plugging values into the formula and solving. The document also discusses using the discriminant to determine how many real solutions an equation has - one, two, or none. Practice problems are given to evaluate discriminants and use them to analyze solutions.
1.5 Solving Inequalities
This document discusses solving inequalities. It defines inequalities and explains that they are relationships where two quantities may not be equal. It describes how to solve inequalities by treating them like equations, except reversing the inequality sign when multiplying or dividing by a negative number. The document also discusses compound inequalities containing "and" or "or" and how to solve them.
4.6 Completing the Square
This document discusses different methods for solving quadratic equations, including finding square roots, solving perfect square trinomials, and completing the square. It provides examples of how to isolate the variable, take square roots, factor perfect square trinomials, add constants to complete the square, and solve quadratic equations using these techniques. Homework problems are assigned from the textbook.
3.3 Systems of Inequalities
This document discusses systems of inequalities and how to solve them by graphing. It explains that a system of inequalities can have many solutions, where a solution satisfies each individual inequality. The key steps are to graph each inequality, find the overlap of the solutions, and use a test point to check the solution set. An example word problem about fundraising ticket sales is provided to demonstrate solving a system of inequalities.
1.2 Data Classification
The document discusses different types and levels of data. There are two types of data: qualitative and quantitative. Qualitative data consists of attributes like names while quantitative data consists of numerical values. There are four levels of measurement for data: nominal, ordinal, interval, and ratio. Each level allows for different statistical calculations and comparisons of the data.
2.3
The document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure, as well as examples of how to find the mean, median, and mode of data sets. It also discusses weighted means, finding the mean of grouped data, and the different shapes distributions can take, such as symmetric, skewed left, and skewed right.
1.3 Algebraic Expressions
1. The order of operations is PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
2. To write an algebraic expression from a word phrase, identify key words that indicate operations and define variables to represent unknown quantities.
3. To evaluate an algebraic expression, substitute values for the variables and simplify using PEMDAS.
4.5 quadratic equations
This document discusses solving quadratic equations by factoring using the zero-product property. It defines key terms like y-intercept, x-intercept, and zero of a function. It explains that to solve a quadratic equation in standard form using factoring, you first factor the quadratic expression, then set each factor equal to 0 and solve for the variable using the zero-product property. Several examples are worked through to demonstrate this process. Homework practice problems are assigned.
4.1 quadratic functions and transformations
This document discusses quadratic functions and their transformations. It defines key terms like parabola, vertex, and axis of symmetry. The vertex form of a quadratic function makes it easy to identify transformations - the a value determines stretching or compression, the h value shifts the graph horizontally, and the k value shifts it vertically. For any quadratic function, the minimum or maximum value will occur at the vertex. Graphing involves plotting the vertex and axis of symmetry, then using a table of values. Writing quadratic functions starts with identifying the vertex coordinates, then using another point to solve for the a value.
Parent night contact&survey
The Triton Travel Club contact survey collects a student's name, address, birthday, and parent/guardian contact information. It asks whether the student will enroll in an upcoming tour and which other countries or places the club should offer trips to in the future, with options including Argentina, Australia, Britain, California, China, Costa Rica, Egypt, France, Germany, Greece, India, Italy, Mexico, New Zealand, Peru, South Africa, Spain, Thailand, Vietnam and others.
7.3 daqy 2
Logarithmic functions are inverses of exponential functions. To graph a logarithmic function:
1. Identify the inverse exponential form.
2. Create a table of values for the exponential form.
3. Invert the ordered pairs.
4. Plot the points and sketch the graph of the logarithm.
Logarithmic functions can be transformed through stretching, compression, reflection, horizontal translation, and vertical translation compared to the parent logarithmic function. Examples are shown to demonstrate how transformations alter the graph.
7.3
This document introduces logarithmic functions as inverses of exponential functions. It defines logarithms as the inverse of an exponential function y = bx, such that y = bx is equivalent to logby = x. The document provides examples of writing exponential equations in logarithmic form and vice versa. It also demonstrates how to evaluate logarithms by using the definition to write them in exponential form and setting the exponents equal. Finally, it defines the common logarithm as a logarithm with base 10, which can be written as logx.
7.1
The document discusses exponential functions of the form f(x) = a*b^x, explaining that they always have a curved shape and asymptote at y=0. It distinguishes between exponential growth, where the value of y increases as x increases, and exponential decay, where the value of y decreases as x increases. Examples are provided to demonstrate how to determine if a function represents growth or decay and to find the y-intercept.
7.2
This document discusses properties and transformations of exponential functions, including stretch, compression, reflection, and horizontal and vertical translation. It also discusses the number e as the base for natural exponential functions and using the function A=Pe^rt to model continuously compounded interest. Examples are provided to demonstrate graphing transformations of exponential functions and using the continuously compounded interest formula.
10.1 part2
This document discusses testing for homogeneity among populations using chi-square tests. It defines homogeneity as populations having the same structure or composition. A test of homogeneity determines if different populations have the same proportions for various categories. It requires using a contingency table and chi-square distribution. An example tests if the same proportion of males and females prefer different pet types using survey data from college students.
10.1 part 1
The document provides an overview of the chi-square distribution and how it can be used for hypothesis testing. It discusses that the chi-square distribution is used to find critical values for determining the area under the curve for a given degrees of freedom. It also gives an example of how chi-square can be used to test if two variables such as keyboard type and time to learn typing are independent.
5.4 synthetic division
This document discusses synthetic division and the remainder theorem. Synthetic division is a process that simplifies long division when dividing a polynomial by a linear factor of the form x - a. It involves setting up the coefficients of the polynomial and multiplying/adding through the process. The remainder theorem states that if a polynomial P(x) is divided by x - a, then the remainder is equal to P(a). It provides a quick way to find the remainder of a polynomial division problem by evaluating the polynomial at the value of a. Examples are given to demonstrate evaluating polynomials using the remainder theorem.
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Ch 8 test review key
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help alleviate symptoms of mental illness and boost overall mental well-being.
6.1
This document discusses roots and radicals. It defines that for every power, there is a corresponding root. It explains that the nth root of a number a is the number that when raised to the nth power equals a. It discusses that if n is odd there is one real nth root, and if n is even and the number is positive there are two real nth roots, and if the number is negative there are no real nth roots. It provides examples of finding cube and fourth roots. It also discusses simplifying radicals and working with variables in radicals.
4.7 The Quadratic Formula
This document discusses solving quadratic equations using the quadratic formula. It begins by reviewing previous methods like factoring and graphing. It then introduces the quadratic formula, which can be used to solve any quadratic equation in standard form. Examples are provided to demonstrate plugging values into the formula and solving. The document also discusses using the discriminant to determine how many real solutions an equation has - one, two, or none. Practice problems are given to evaluate discriminants and use them to analyze solutions.
1.5 Solving Inequalities
This document discusses solving inequalities. It defines inequalities and explains that they are relationships where two quantities may not be equal. It describes how to solve inequalities by treating them like equations, except reversing the inequality sign when multiplying or dividing by a negative number. The document also discusses compound inequalities containing "and" or "or" and how to solve them.
4.6 Completing the Square
This document discusses different methods for solving quadratic equations, including finding square roots, solving perfect square trinomials, and completing the square. It provides examples of how to isolate the variable, take square roots, factor perfect square trinomials, add constants to complete the square, and solve quadratic equations using these techniques. Homework problems are assigned from the textbook.
3.3 Systems of Inequalities
This document discusses systems of inequalities and how to solve them by graphing. It explains that a system of inequalities can have many solutions, where a solution satisfies each individual inequality. The key steps are to graph each inequality, find the overlap of the solutions, and use a test point to check the solution set. An example word problem about fundraising ticket sales is provided to demonstrate solving a system of inequalities.
1.2 Data Classification
The document discusses different types and levels of data. There are two types of data: qualitative and quantitative. Qualitative data consists of attributes like names while quantitative data consists of numerical values. There are four levels of measurement for data: nominal, ordinal, interval, and ratio. Each level allows for different statistical calculations and comparisons of the data.
2.3
The document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure, as well as examples of how to find the mean, median, and mode of data sets. It also discusses weighted means, finding the mean of grouped data, and the different shapes distributions can take, such as symmetric, skewed left, and skewed right.
1.3 Algebraic Expressions
1. The order of operations is PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
2. To write an algebraic expression from a word phrase, identify key words that indicate operations and define variables to represent unknown quantities.
3. To evaluate an algebraic expression, substitute values for the variables and simplify using PEMDAS.
4.5 quadratic equations
This document discusses solving quadratic equations by factoring using the zero-product property. It defines key terms like y-intercept, x-intercept, and zero of a function. It explains that to solve a quadratic equation in standard form using factoring, you first factor the quadratic expression, then set each factor equal to 0 and solve for the variable using the zero-product property. Several examples are worked through to demonstrate this process. Homework practice problems are assigned.
4.1 quadratic functions and transformations
This document discusses quadratic functions and their transformations. It defines key terms like parabola, vertex, and axis of symmetry. The vertex form of a quadratic function makes it easy to identify transformations - the a value determines stretching or compression, the h value shifts the graph horizontally, and the k value shifts it vertically. For any quadratic function, the minimum or maximum value will occur at the vertex. Graphing involves plotting the vertex and axis of symmetry, then using a table of values. Writing quadratic functions starts with identifying the vertex coordinates, then using another point to solve for the a value.
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Parent night contact&survey
The Triton Travel Club contact survey collects a student's name, address, birthday, and parent/guardian contact information. It asks whether the student will enroll in an upcoming tour and which other countries or places the club should offer trips to in the future, with options including Argentina, Australia, Britain, California, China, Costa Rica, Egypt, France, Germany, Greece, India, Italy, Mexico, New Zealand, Peru, South Africa, Spain, Thailand, Vietnam and others.
7.3 daqy 2
Logarithmic functions are inverses of exponential functions. To graph a logarithmic function:
1. Identify the inverse exponential form.
2. Create a table of values for the exponential form.
3. Invert the ordered pairs.
4. Plot the points and sketch the graph of the logarithm.
Logarithmic functions can be transformed through stretching, compression, reflection, horizontal translation, and vertical translation compared to the parent logarithmic function. Examples are shown to demonstrate how transformations alter the graph.
7.3
This document introduces logarithmic functions as inverses of exponential functions. It defines logarithms as the inverse of an exponential function y = bx, such that y = bx is equivalent to logby = x. The document provides examples of writing exponential equations in logarithmic form and vice versa. It also demonstrates how to evaluate logarithms by using the definition to write them in exponential form and setting the exponents equal. Finally, it defines the common logarithm as a logarithm with base 10, which can be written as logx.
7.1
The document discusses exponential functions of the form f(x) = a*b^x, explaining that they always have a curved shape and asymptote at y=0. It distinguishes between exponential growth, where the value of y increases as x increases, and exponential decay, where the value of y decreases as x increases. Examples are provided to demonstrate how to determine if a function represents growth or decay and to find the y-intercept.
7.2
This document discusses properties and transformations of exponential functions, including stretch, compression, reflection, and horizontal and vertical translation. It also discusses the number e as the base for natural exponential functions and using the function A=Pe^rt to model continuously compounded interest. Examples are provided to demonstrate graphing transformations of exponential functions and using the continuously compounded interest formula.
10.1 part2
This document discusses testing for homogeneity among populations using chi-square tests. It defines homogeneity as populations having the same structure or composition. A test of homogeneity determines if different populations have the same proportions for various categories. It requires using a contingency table and chi-square distribution. An example tests if the same proportion of males and females prefer different pet types using survey data from college students.
10.1 part 1
The document provides an overview of the chi-square distribution and how it can be used for hypothesis testing. It discusses that the chi-square distribution is used to find critical values for determining the area under the curve for a given degrees of freedom. It also gives an example of how chi-square can be used to test if two variables such as keyboard type and time to learn typing are independent.
5.4 synthetic division
This document discusses synthetic division and the remainder theorem. Synthetic division is a process that simplifies long division when dividing a polynomial by a linear factor of the form x - a. It involves setting up the coefficients of the polynomial and multiplying/adding through the process. The remainder theorem states that if a polynomial P(x) is divided by x - a, then the remainder is equal to P(a). It provides a quick way to find the remainder of a polynomial division problem by evaluating the polynomial at the value of a. Examples are given to demonstrate evaluating polynomials using the remainder theorem.
5.4 long division
Long division can be used to divide polynomials in a similar way to dividing numbers. The key steps are to set up the division problem, divide the term of the dividend by the term of the divisor, multiply the divisor by the quotient term and subtract, then bring down the next term of the dividend and repeat. This polynomial long division allows polynomials to be factored by finding all divisor polynomials that give a remainder of zero. The factor theorem can also be used to check if a linear polynomial is a factor by setting it equal to zero and checking if it makes the other polynomial equal to zero.
5.3
This document discusses solving polynomial equations by factoring. It provides examples of factoring polynomials, including factoring the difference and sum of cubes. Factoring by substitution is also introduced as a method for factoring polynomials of degree 4 or higher. The document demonstrates solving polynomial equations by factoring the expressions and setting each factor equal to 0. Both real and imaginary solutions may be obtained depending on whether the factors are real or complex numbers. Graphing is presented as an alternative method to find real solutions of a polynomial equation.
9.3 Part 1
The document discusses inferences for correlation and regression. It provides an example of testing the correlation between percentage of population with a college degree (x) and percentage growth in income (y) for 6 Ohio communities. There is a positive correlation between x and y, but this does not necessarily mean higher education causes higher earnings. The document also discusses measuring the spread of data points around the least squares line, including the standard error of estimate, using an example of how much copper sulfate dissolves in water at different temperatures.
9.1 9.2 9.3 using the graph calc
This document provides guided notes on inferences for correlation and regression. It discusses how the sample correlation coefficient and least squares line estimate population parameters and require assumptions about the data. It also outlines how to test the population correlation coefficient using a significance test and interpret the results. An example is provided testing the correlation between education levels and income growth. Students are asked to practice computing the standard error of estimate from a data set and answering summary questions.
5.2
1. The document discusses writing polynomials in factored form and finding the zeros of polynomial functions. It defines linear factors, roots, zeros, and x-intercepts as equivalent terms.
2. Examples are provided of writing polynomials in factored form using the factor theorem to find the zeros, and then graphing the polynomial function based on its zeros.
3. The factor theorem states that a linear expression x - a is a factor of a polynomial if and only if a is a zero of the related polynomial function. This allows writing a polynomial given its zeros.
5.1 part 2
This document discusses how to describe the shape of a cubic function by listing it in standard form, describing the end behavior of the graph, determining the possible number of turning points using a table of values, and determining the increasing and decreasing intervals. It explains that to describe the shape, you identify the sign of the leading coefficient to determine the end behavior and the number of turning points, which is one less than the possible degree. The document also discusses using differences of consecutive y-values in a table to determine the least degree of the polynomial function that could generate the data, with constant first differences indicating linear, constant second differences indicating quadratic, and constant third differences indicating cubic.
5.1[1]
This document defines key concepts related to polynomials and polynomial functions. It defines monomials as terms involving variables and exponents, and polynomials as sums of monomials. The degree of a polynomial is the highest exponent among its terms. Polynomial functions are polynomials written in terms of a single variable. Standard form arranges polynomial terms by descending degree. Polynomials are classified by degree and number of terms. Higher degree polynomials can have more turning points and their end behavior depends on the leading term. Examples show determining standard form, classifying polynomials, identifying end behavior and increasing/decreasing parts of graphs.
9.1
This document discusses scatter diagrams and linear correlation. It provides examples of scatter diagrams that do and do not show linear correlation. It defines the correlation coefficient r as a measure of linear correlation between two variables on a scatter plot, with values between -1 and 1. It presents formulas for calculating r and provides an example of computing r using wind velocity and sand drift rate data. It cautions that correlation does not necessarily imply causation and that lurking variables can influence the correlation between two variables.
9.2 lin reg coeff of det
1) Linear regression finds the "best-fitting" linear relationship between two variables by minimizing the vertical distances between the data points and the linear equation line.
2) The coefficient of determination, r^2, measures how well the linear relationship described by the regression line fits the actual data, with higher r^2 values indicating less unexplained variability.
3) r^2 has an interpretation as the percentage of the total variation in the response variable that is explained by the explanatory variable.