4.3
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This document discusses modeling quadratic functions and writing equations of parabolas. It explains that the vertex form can be used when the vertex and a point are known, and standard form can be used with any three points on the parabola. Examples are provided on finding the standard form equation of a parabola passing through three given points. The document concludes with an assignment to complete practice problems 7 through 16, starting with problem 16 first.
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AnaylticaltsJ'ette Novakovich
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The document provides guidance on writing effective analytical topic sentences when analyzing a text. It advises to avoid merely descriptive topic sentences and instead make a claim about the text being analyzed. A good analytical topic sentence should: 1) Make an interpretive claim about some aspect of the text; 2) Signal where the paragraph is going to take the argument; and 3) Echo and support the overall thesis statement. Sample topic sentences are provided and analyzed to demonstrate how they incorporate these elements of an effective analytical topic sentence.8.2 p value guided notes
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This document provides guidance on performing hypothesis tests of the mean. It outlines the procedures for testing the mean when the population standard deviation is known or unknown. An example is provided for each case, testing if the mean number of sunspots or drug remission time is different than a hypothesized value. Key steps are calculating the test statistic, determining the p-value, and comparing it to the significance level to determine whether to reject or fail to reject the null hypothesis. Questions at the end review the procedures and concepts covered.2.6
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Families of functions share common characteristics, with a parent function as the simplest form. Transformations can create other functions in the family. One type is translation, which shifts the graph horizontally and/or vertically without changing shape. Reflection flips the graph over an axis, with points the same distance from the axis of reflection. Vertical stretch multiplies y-values by a factor greater than 1, while vertical compression uses a factor between 0 and 1. Combining transformations, like reflecting and compressing, results in a new function rule.9.1 9.2 9.3 using the graph calc
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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.7.4 part 2
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The document discusses constructing confidence intervals for the difference between two population parameters using independent random samples from two populations. There are several types of confidence intervals that can be constructed, including for the difference between two means (亮1 - 亮2) when the standard deviations are known or unknown, and for the difference between two proportions (p1 - p2). Examples are provided to illustrate how to construct confidence intervals for the difference between means and proportions.2.2
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Direct variation functions have a constant ratio between the input (x) and output (y) values. The formula for a direct variation is y=kx, where k is the constant of variation. To determine if a relationship is direct variation:
1) Check if the ratio of all y/x pairs is the same
2) Simplify y/x to find k
3) The direct variation equation is y=kx
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.2.7
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This document discusses absolute value functions and their graphs. It defines the parent graph of an absolute value function as f(x)=x and describes its V-shape and axis of symmetry. Examples are given of absolute value functions after various transformations from the parent graph, along with the general form y=a(x-h)+k. The document also describes how to identify the vertex and axis of symmetry from the equation of an absolute value function and how to write the equation of an absolute value function given its graph. Homework problems are assigned from the textbook.8.4
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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.Parent night contact&survey
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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
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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
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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
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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
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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.10.2
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This document discusses the chi-square goodness of fit test, which is used to check if observed data counts match the expected distribution of counts into categories. It examines whether a population follows a specified theoretical distribution.10.1 part2
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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- 1. 4.3 MODELING WITH QUADRATIC FUNCTIONS
- 2. WRITING THE EQUATION OF A PARABOLA When you know the vertex and a point on a parabola you can write the equation of the parabola using vertex form. If you do not know the vertex, then you can use any three points on the parabola and standard form to write the equation of the parabola.
- 3. WRITING THE EQUATION OF A PARABOLA
- 4. EXAMPLE: FIND AN EQUATION IN STANDARD FORM OF THE PARABOLA PASSING THROUGH THE POINTS
- 5. EXAMPLE: FIND AN EQUATION IN STANDARD FORM OF THE PARABOLA PASSING THROUGH THE POINTS
- 6. EXAMPLE: FIND AN EQUATION IN STANDARD FORM OF THE PARABOLA PASSING THROUGH THE POINTS
- 7. Page 210 COMPARING QUADRATIC MODELS
- 8. ASSIGNMENT Page 212 #7 16 all Do #16 first!