Exponents Review
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This document discusses exponent laws and properties for simplifying exponential expressions including the product of powers rule, power to a power rule, power of a product rule, anything to the zero power is 1 rule, and quotient of powers rule. It explains how to combine or simplify exponential terms with the same base using multiplication and division properties of exponents.
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Parent functions are families of graphs that share unique properties. Transformations can move the graph around the plane. The main parent functions explored are the constant, linear, absolute value, quadratic, cubic, square root, cubic root, and exponential functions. Each has a characteristic shape and number of intercepts. Domains and ranges depend on the specific function but often extend to positive and negative infinity.
Module 2 Lesson 2 Notes
This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.
Module 1 Lesson 1 Remediation Notes
The document provides information about functions and relations. It defines a function as a relation where each x-value is paired with exactly one y-value. To determine if a relation is a function, it describes using the vertical line test, where a relation is a function if a vertical line can only intersect the graph at one point. It gives examples of applying the vertical line test to graphs and determining the domain and range of relations.
M1 L5 Remediation Notes
This document discusses solving systems of equations and inequalities through three main methods: graphing, substitution, and elimination. It provides examples of each method. For graphing systems, it explains the three possibilities for the graphs: consistent systems with one solution where the lines intersect, inconsistent systems with no solution where the lines are parallel, and dependent systems with infinite solutions where the lines coincide. It then works through examples of using substitution and elimination to solve systems algebraically. [/SUMMARY]
Relations and Functions
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
Multiplying Polynomials (no conjugates)
This document discusses techniques for multiplying polynomials including FOIL, the double distributive property, and squaring binomials. It provides examples of squaring binomial expressions like (x + 3)2 by multiplying (x + 3) by itself using FOIL. The document also lists more example problems but does not show the solutions.
Fractions, Decimals, and Percents
This document defines decimals, fractions, and percents and provides steps for converting between them. Decimals are numbers with a decimal point, fractions show parts of a whole, and percents express amounts out of 100. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a fraction to a percent, change it to a decimal then multiply by 100. Converting between other forms follows similar steps of changing the number to an equivalent decimal or percent value.
C3 Study Slides - MAT 151
This document contains lecture slides about statistics for describing, exploring, and comparing data. It discusses measures of center such as the mean, median, and mode. It also discusses variance and standard deviation as measures of spread. Additional topics covered include finding the mode, determining if a value is unusually high or low based on the mean and standard deviation, calculating percentiles, and comparing the detail provided by different graphic displays of data.
C2 Study Slides - MAT 151
The document contains multiple choice questions about summarizing and graphing data. It asks questions about outliers, frequency distributions, class boundaries, histograms, and types of charts including pie charts. One question asks the reader to use a pie chart showing housing types to find the number of people living in single family housing in a town of 12,200 people.
C1 Study Slides - MAT151
The document contains lecture slides on introductory statistics topics including definitions of population, quantitative vs qualitative data, types of measurement scales, and examples of different sampling methods and study designs. Key points covered are the definition of a population as the complete collection of all elements, examples of quantitative data like weights vs qualitative nominal categories, ordinal scales involving ranking, and retrospective study designs using existing historical data.
C3 test Doc
This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalences, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
C3 test
This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalence, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
Intro to Logs
The document introduces logarithms, defining them as the exponent that a fixed number (the base) must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and covers the key properties of logarithms including product, quotient, power, expanding, condensing, and the change of base formula.
Logs
The document introduces logarithms, defining them as the exponent that a base number must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and discusses properties of logarithms such as the product, quotient, and power properties. It also covers expanding, condensing, and changing the base of logarithmic expressions.
Intro to Polynomials
This document discusses polynomial functions and how to graph them. It defines a polynomial as a sum of terms with non-negative integer exponents. Polynomial graphs are smooth curves that may be lines, parabolas, or higher-order curves. To graph a polynomial, one determines the end behavior from the leading term, finds the x-intercepts by setting the polynomial equal to 0, and uses intercepts and test points to plot the graph over intervals. Multiplicity of roots affects whether the graph crosses or is tangent to the x-axis at those points.
Completing the Square
The document explains how to solve quadratic equations by completing the square. It defines a perfect square trinomial as having the form x^2 + bx + c, where c is the square of half of b. It provides steps for completing the square, which involves adding a constant term to both sides of the equation such that the left side becomes a perfect square trinomial that can be factorized. This process results in the solution(s) to the quadratic equation. Two examples demonstrating this process are included.
Graph Quadratics
The document discusses graphing quadratic functions in standard form (y=ax^2 + bx + c). It explains that the graph is a parabola that can open up or down depending on whether a is positive or negative. The line of symmetry for the parabola passes through the vertex and is given by the equation x=-b/2a. The steps to graph are: 1) find the line of symmetry, 2) plug the x-value into the original equation to find the vertex, 3) find two other points and reflect them across the line of symmetry.
Composite functions
Composite functions are formed by taking the output of one function and using it as the input of another function. This is shown notationally as f(g(x)), where the result of g(x) is used as the input for f. Changing the order of the functions changes the result, as the output of the inner function determines the input to the outer function. Examples show evaluating composite functions by substituting the output of the inner function into the outer function and simplifying.
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Exponents Review
- 2. Basic Terms BASE EXPONENT means Important Examples
- 3. IMPORTANT EXAMPLES Variable Examples
- 5. MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base.
- 6. MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base.
- 7. MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions.
- 8. ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1 .
- 9. DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.
- 10. DIVISION PROPERTIES POWER OF A QUOTIENT