5.1 part 1
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This document summarizes key concepts about polynomial functions including: - Definitions of monomials, polynomials, and polynomial functions - The standard form and classification of polynomials by degree and number of terms - How the degree of a polynomial affects the shape of its graph, including the number of turning points and end behavior - Examples of determining the end behavior and classifying increasing and decreasing parts of polynomial graphs
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5.1[1]![5.1[1]](https://cdn.slidesharecdn.com/ss_thumbnails/5-11-121219075353-phpapp01-thumbnail.jpg?width=560&fit=bounds)
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5.4 synthetic divisionleblanceThis 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
5.4 long divisionleblanceLong 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
5.3leblanceThis 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
9.3 Part 1leblanceThe 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
9.1 9.2 9.3 using the graph calcleblanceThis 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.
9.1
9.1leblanceThis 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
9.2 lin reg coeff of detleblance1) 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.
8.4
8.4leblanceMany 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.
8.3 critical region
8.3 critical regionleblanceThe 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%.
10.3 part 1leblance
5.1 part 1
- 1. Chapter 5: Polynomials and Polynomial Functions 5.1: Polynomial Functions
- 2. Definitions A monomial is a real number, a variable, or a product of a real number and one or more variables with whole number exponents. Examples: The degree of a monomial in one variable is the exponent of the variable.
- 3. Definitions A polynomial is a monomial or a sum of monomials. Example: The degree of a polynomial in one variable is the greatest degree among its monomial terms. Example:
- 4. Definitions A polynomial function is a polynomial of the variable x . A polynomial function has distinguishing “behaviors” The algebraic form tells us about the graph The graph tells us about the algebraic form
- 5. Definitions The standard form of a polynomial function arranges the terms by degree in descending order Example:
- 6. Definitions Polynomials are classified by degree and number of terms. Polynomials of degrees zero through five have specific names and polynomials with one through three terms also have specific names. Degree Name 0 Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic Number of Terms Name 1 Monomial 2 Binomial 3 Trinomial 4+ Polynomial with ___ terms
- 7. Example Write each polynomial in standard form. Then classify it by degree and by number of terms.
- 8. Example Write each polynomial in standard form. Then classify it by degree and by number of terms.
- 9. Polynomial Behavior The degree of a polynomial function Affects the shape of its graph Determines the number of turning points (places where the graph changes direction) Affects the end behavior (the directions of the graph to the far left and to the far right)
- 10. Polynomial Behavior The graph of a polynomial function of degree n has at most n – 1 turning points. Odd Degree = even number of turning points Even Degree = odd number of turning points Think about this: If a polynomial has degree 2, how many turning points can it have? If a polynomial has degree 3, how many turning points can it have?
- 11. Polynomial Behavior End behavior is determined by the leading term
- 13. Example Determine the end behavior of the graph of each polynomial function.
- 14. Example Determine the end behavior of the graph of each polynomial function.
- 15. Increasing and Decreasing Remember: We read from left to right! A function is increasing when the y-values increase as the x-values increase A function is decreasing when the y-values decrease as the x-values increase
- 16. Example: Identify the parts of the graph that are increasing or decreasing
- 17. Example: Identify the parts of the graph that are increasing or decreasing
- 18. Homework P285 #8 – 31all