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This document discusses linear equations and their various forms. It covers finding the equation of a line given a point and slope using point-slope form, or given two points using the formula for slope and point-slope form. It also discusses writing equations in standard and slope-intercept form, finding intercepts to graph lines, and writing equations of lines parallel or perpendicular to given lines through a specified point. Examples are provided for each concept.
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Direct variation graphs always pass through the origin and have a constant slope equal to the constant of variation. They can be graphed by making a table of values and plotting points or using the slope and origin. Direct variations can be used to solve word problems by setting up proportions.
8.4
Many statistical tests use paired data samples to compare two population means. Paired data occurs naturally in "before and after" situations where the same item is measured before and after a treatment. When testing paired data, the proper procedure is to run a one-sample test on the differences between each pair of measurements. This allows researchers to determine if a treatment had a statistically significant effect based on the average difference between paired measurements.
3.3 percentiles and boxandwhisker plot
The document defines percentiles and how they divide data into hundredths, with quartiles specifically dividing data into fourths. It describes how to compute quartiles and the interquartile range. Finally, it explains how to create and interpret box-and-whisker plots using the five number summary to visually depict the spread and symmetry of a data set.
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.
4.1 quadratic functions and transformations
The document discusses quadratic functions and transformations. It defines key terms like parabola, vertex, axis of symmetry, minimum and maximum values. It explains how the coefficients in the vertex form of a quadratic function relate to transformations - the a value determines if the parabola is vertically stretched (larger a) or compressed (smaller a), and a negative a reflects the graph over the x-axis. Examples are given of graphing and writing quadratic functions based on transformations from the standard form.
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.
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2.2 bar graphs, circle graphs, and time series graphs
2.2 bar graphs, circle graphs, and time series graphs
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2.3
This document discusses linear functions and slope-intercept form. It defines slope and describes how to calculate the slope between two points. It explains that a linear function is one whose graph is a straight line and can be represented by a linear equation. It defines y-intercepts and x-intercepts as the points where a line crosses the y-axis and x-axis. It shows how to write a linear equation in slope-intercept form using the slope and y-intercept. Examples are given for finding the slope and y-intercept of a line and writing the line's equation in slope-intercept form. It also demonstrates how to graph a line by using its equation in slope-intercept form.
3.5 write and graph equations of lines
This geometry document covers writing equations of lines in slope-intercept form. It reviews the concepts of slope and y-intercept and teaches that an equation of a line needs a slope and y-intercept. Examples are given of writing equations given these components. The document also addresses writing equations of lines parallel or perpendicular to given lines and passing through given points. Students are provided practice problems to write equations of lines in different scenarios.
Geometry unit 3.8
The document discusses parallel and perpendicular lines. It provides examples of identifying parallel and perpendicular lines based on their slopes. It also discusses writing equations of lines parallel and perpendicular to given lines, and applications involving parallel and perpendicular lines in geometry problems involving shapes like parallelograms, rectangles, and right triangles.
5.1 writing linear equations day 1
This chapter discusses writing linear equations in slope-intercept form (y=mx+b) by either knowing the slope and y-intercept or by being given two points on the line. It provides examples of writing equations of lines from graphical representations or when given the slope and y-intercept numerically. Students are guided in writing linear equations from different scenarios like being given two points or the cost equation of a recording studio based on hours of studio time.
5.5 parallel perp lines
The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
5.4 write linear equations in standard form day 1
This document provides instructions on writing linear equations in standard form. It defines the standard form as Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. Examples are provided of writing equivalent equations in standard form by multiplying both sides of an equation by a constant. The document also demonstrates writing the equation of a line given its slope and a point, or given just a single point. Students are guided in writing equations of horizontal and vertical lines and completing partial equations where a coefficient is missing. Practice problems are assigned for writing equations in standard form.
Unit 6.1
The document provides examples for writing linear equations in slope-intercept form (y=mx+b) given different representations of a line:
1) Given the slope (m) and y-intercept (b), write the equation.
2) Given a graph of the line, calculate the slope and y-intercept to write the equation.
3) Given the coordinates of two points on the line, calculate the slope and use it to write the equation.
4) Given an equation and a point that the line passes through, solve for the missing slope (m) or y-intercept (b) value.
2.3 and 2.4 Lines
This document discusses linear functions and different forms of writing linear equations. It introduces linear functions and their representation as linear equations. It describes intercepts as the points where a line crosses the x-axis or y-axis. The slope-intercept form of a line is given as y = mx + b, where m is the slope and b is the y-intercept. It provides examples of writing equations in slope-intercept form and finding the slope and y-intercept. It also covers the point-slope form and how to write the equation of a line given a point and slope or two points. Finally, it discusses standard form for linear equations.
5.5 writing linear equations
1) Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other.
2) To determine if two lines are parallel or perpendicular, you calculate their slopes. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals, the lines are perpendicular.
3) To find the equation of a line parallel or perpendicular to a given line, you use the slope of the given line and the point-slope formula. For a parallel line, you use the same slope. For a perpendicular line, you use the negative reciprocal slope.
5.5 parallel and perpendicular lines (equations) day 1
The document discusses writing equations of parallel and perpendicular lines, including identifying whether lines are parallel or perpendicular based on their slopes. It provides examples of writing equations of parallel and perpendicular lines given a point and the slope or line of another line. Guidance problems are also included to practice these skills.
Cal 3
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
Copy_of_slopeofaline (1).ppt
- A line is defined as a set of points on a plane where the slope between any two points is always equal. A line on the Cartesian plane can be described by a linear equation.
- The slope of a line describes its direction, and can be calculated using the rise over run formula. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.
- The equation of a line can be written in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. The slope and y-intercept can be used to write the equation of a line given a point and the slope, or two points on the line.
Copy_of_slopeofaline.ppt
- A line is defined as a set of points on a plane where the slope between any two points is always equal. A line on the Cartesian plane can be described by a linear equation.
- The slope of a line describes its direction, and can be calculated using the rise over run formula. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.
- The equation of a line can be written in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. The slope and y-intercept can be used to write the equation of a line given a point and the slope, or two points on the line.
Copy_of_slopeofaline.ppt
- A line is defined as a set of points on a plane where the slope between any two points is equal. A line on the Cartesian plane can be described by a linear equation.
- The slope of a line describes its direction, and can be calculated using the rise over run formula. Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.
- The equation of a line can be written in various forms, including slope-intercept form where y = mx + b, with m being the slope and b being the y-intercept.
Copy_of_slopeofaline.ppt
Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A line Slope of A lineSlope of A line Slope of A line Slope of A line
Formulas
The document discusses formulas for calculating distance, midpoints, and slopes of lines on a coordinate plane. It defines key terms like x-axis, y-axis, origin, and introduces the distance, midpoint, and slope formulas. Examples are provided to demonstrate calculating distances and slopes between points and finding midpoints, and describing lines based on whether their slopes are positive, negative, undefined, or zero.
2.2 linear equations and 2.3 Slope
The document discusses linear equations and slope. It covers plotting points on a coordinate plane, calculating slope using the rise over run formula, writing equations in slope-intercept form, finding the x- and y-intercepts, and graphing lines by making a table or using the slope and y-intercept. Methods are provided for determining the equation of a line given two points, the slope and one point, or from a graph.
Integrated Math 2 Section 8-1
The document discusses parallel and perpendicular lines, explaining that parallel lines have the same slope while perpendicular lines have slopes that are negative reciprocals. It provides examples of finding the slope of parallel and perpendicular lines and writing equations of lines given properties like a point and slope. Formulas for slope-intercept and point-slope forms of linear equations are also presented.
Int Math 2 Section 8-1 1011
The document discusses key concepts about parallel and perpendicular lines including how to determine if two lines are parallel or perpendicular based on their slopes, and how to write equations of parallel and perpendicular lines. It provides examples of finding slopes of parallel and perpendicular lines and writing equations to represent lines given certain conditions. Formulas for slope-intercept and point-slope forms are also presented.
Math14 lesson 5
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
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5.5 parallel and perpendicular lines (equations) day 1
5.5 parallel and perpendicular lines (equations) day 1
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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.
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.
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[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.
9.2 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.
8.3 critical region
The document describes testing a claim about the proportion of seeds from a new hybrid wheat variety that germinate. A botanist claimed the proportion for the hybrid that germinate is 80%, the same as the parent plants. An experiment was conducted where 400 seeds from the hybrid were tested, and 312 germinated. The summary provides instructions to use a 5% level of significance and critical regions to test the claim that the proportion germinating for the hybrid is 80%.
8.3 critical region
Testing a proportion using critical regions follows a similar process to testing a mean. The main difference is that a proportion represents a probability rather than a measurement. When using critical regions to test a proportion, if the sample test statistic falls within the critical region, the null hypothesis is rejected, and if it falls outside the critical region, the null hypothesis is not rejected. The test statistic is the z-score calculated from the sample proportion and null hypothesized proportion.
8.4 guided notes
1. Paired data involves dependent samples that are measured in pairs, such as measurements taken from the same subject before and after a treatment. When comparing paired data, the proper test is a one-sample t-test on the differences between pairs.
2. The null hypothesis for a paired t-test states that the mean of the differences between pairs is equal to zero, indicating no change between measurements. The alternative hypothesis depends on the specific problem but can be left-tailed, right-tailed, or two-tailed.
3. The key steps in a paired t-test are calculating the differences between pairs, finding the mean and standard deviation of the differences, determining the t-statistic, computing the
8.3 p value
1) Testing a proportion uses a binomial distribution with hypotheses about p, the probability of success on each trial. The test statistic is calculated and compared to a normal distribution to get a p-value.
2) An example tests whether a new eye surgery technique is better than the old technique based on a trial with 225 surgeries and 88 successes, using a 1% significance level.
3) Key steps are to check conditions, calculate the test statistic, find the p-value using the normal distribution, and either reject or fail to reject the null hypothesis based on the significance level.
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2.4
- 1. 2.4 MORE ABOUT LINEAR EQUATIONS
- 3. POINT – SLOPE FORM
- 4. EXAMPLE: WRITING AN EQUATION GIVEN A POINT AND THE SLOPE 3 A line passes through ( −5, 2 ) with slope 5 What is an equation of the line?
- 5. EXAMPLE: WRITING AN EQUATION GIVEN A POINT AND THE SLOPE A line passes through ( 7, −1) with slope −3 What is an equation of the line?
- 6. WRITING AN EQUATION GIVEN TWO POINTS y2 − y1 1. Find the slope m= x2 − x1 2. Substitute the slope and one point into the point-slope form y − y1 = m ( x − x1 )
- 7. EXAMPLE A line passes through (3, 2) and (5, 8). What is an equation of the line in point-slope form?
- 8. EXAMPLE A line passes through (- 5, 0) and (0, 7). What is an equation of the line in point-slope form?
- 10. WRITING AN EQUATION IN STANDARD FORM WITH INTEGER COEFFICIENTS
- 11. EXAMPLE: WRITE THE EQUATION IN STANDARD FORM. USE INTEGER COEFFICIENTS. 3 y = x −5 4
- 13. GRAPHING AN EQUATION The quickest way to graph a line is to: 1. Find the x – intercept by setting y = 0 2. Find the y – intercept by setting x = 0 3. Plot the intercepts 4. Draw the line
- 14. EXAMPLE: WHAT ARE THE INTERCEPTS OF THE EQUATION? GRAPH THE EQUATION. 3 x + 5 y = 15
- 16. WRITING EQUATIONS OF PARALLEL AND PERPENDICULAR LINES 1. Find the appropriate slope 2. Use the given point and point – slope form to write the equation of the line
- 17. EXAMPLE: WRITE THE EQUATION OF THE LINE IN SLOPE – INTERCEPT FORM. The line parallel to y = 6 x − 2 and through ( 1, −3)
- 18. EXAMPLE: WRITE THE EQUATION OF THE LINE IN SLOPE – INTERCEPT FORM. 2 The line perpendicular to y = −4 x + 3 and through ( 8,5 )
- 19. EXAMPLE: WRITE THE EQUATION OF THE LINE IN SLOPE – INTERCEPT FORM. The line parallel to 4 x + 2 y = 7 and through ( 4, −2 )
- 20. EXAMPLE: WRITE THE EQUATION OF THE LINE IN SLOPE – INTERCEPT FORM. 2 The line perpendicular to y = x − 1 3 and through ( 0,6 )
- 21. HOMEWORK Page 86 #6 – 8 all, 11 – 29 odd, 32 – 35 all, 42 – 44 all