Embedded Files
NOTE: Resources are listed in descending order (the list for semester 1 starts at the bottom of the page). Sorry for any inconvenience.
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(31) Record your answers to the MCAS Practice Test - April 2015 in the Google Form below:
Google Form for MCAS Practice Test - April 2015
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(30) If you need the definition of a math concept/ skill or formula (multilingual) and/ or to see a basic example using that concept if applicable, you may want to check out the online math glossary from McGraw Hill Mathematics (see link below):
http://www.glencoe.com/apps/eGlossary612/landing.php
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(29) Simplifying complex fractions
Here is an example of using the "Granddaddy LCD" (the LCD of all the denominators) for simplifying complex fractions:
Using the "Grandaddy LCD" for simplifying numerical fractions
Using the "Grandaddy LCD" to simplify complex fractions (the last 2 minutes of this video are the most important for the strategy using the "Granddaddy LCD", you may want to watch that part very carefully).
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(28) Solving rational equations algebraically (see i and ii below) and graphically (see iii)
(i) Here is an example of solving a rational equation using cross multiplication (when we have one fraction in simplest form on each side of the equation):
Using cross multiplication to solve rational equations
NOTE: The "set of restrictions" mentioned in this video is our domain. To factor the quadratic equation that we have to eventually solve in this problem, you may use GCF and reverse FOIL as shown in the video OR GCF and Box.
(ii) Here is an example of solving a rational equation using the "Grandaddy LCD" (when we don't have a simple proportion; the goal is to get rid of all the denominators by using the "Grandaddy LCD" to multiply by on both sides of the equation):
Using Grandaddy LCD to solve a rational equation
NOTE: In this video, they used the check as a plug in the original equation instead of the domain. Please check your classroom notes for finding domain and use that as an easier, more convenient "check". Although the author of this video recommends using LCD only if the denominators have common factors, keep in mind that the LCD method is the general strategy for solving rational equations regardless of the GCFs of the denominators.
(iii) Here is a video presenting the algebraic solutions along with the graphical solutions using a graphing calculator (the graphs are used as a check of the algebraic work ) - click the link below and watch Ex 2: Solve Rational Equations Algebraically by Clearing Fractions and Solve Graphically:
Solving rational equations algebraically and graphically
NOTE: Both problems presented in Ex 2 could have been solved using cross multiplication for the algebraic work!
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(27) Multiplying and dividing rational expressions
For extra help/ extra examples on multiplying and dividing rational expressions, check the links below:
Multiplying Rational Expressions
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(26) Adding and subtracting rational expressions
For extra help/ extra examples on adding and subtracting rational expressions, check the links below:
Adding and Subtracting Rational Expressions
Adding and Subtracting Rational Expressions (more examples)
Adding and Subtracting Rational Expressions (some more examples)
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(25) Solving exponential and logarithmic equations
Solving Exponential and Logarithmic Equations from YayMath!
Solving Type II Exponential Equations from Khan Academy
More Challenging (Pre- AP) Exponential and Logarithmic Equations
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(24) Graphing a logarithmic function
For extra help/ extra examples on graphing a logarithmic function using transformations from the parent function, check one of the links below:
Graphing a logarithmic function (basic example using numerical table first)
Graphing transformations of logarithmic functions (various examples)
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(23) Finding the inverse of a function
If you need help/ extra practice on finding the inverse of a function algebraically, you may want to check out the following video:
Step- by- step algorithm on finding the inverse of a function (various examples)
NOTE: You shouldn't write the equation of the inverse function in standard form if by doing that you would not be able to see the transformations easily anymore!
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(22) Properties of logarithms
If you need help/ extra practice on using properties of logarithms to evaluate logarithms, expand, condense or approximate logarithmic expressions, you may want to check some of the examples found in the list below (the link will show a collection of YouTube videos that you can pick from to reinforce what you learned in class.
Properties of Logarithms (collection of videos)
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(21) Finding the domain of a function algebraically
If you need help/ extra practice examples on finding the domain of various functions algebraically, you may want to check out the following video:
Finding the domain algebraically (various functions)
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(20) Finding the range of a function algebraically
If you need help/ extra practice examples on finding the range of various functions algebraically, you may want to check out the following video:
Finding the range algebraically (various functions)
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(19) If you need extra practice on topics that we already discussed and/ or new topics that we are currently learning, you may want to check out the Math Worksheets Go! website
They have free printable math worksheets with answer keys included, and free practice problems that you can try online, and get immediate feedback and/ or hints on the solutions to the problems.
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(18) Solving exponential applications algebraically using logarithms.
(i) Finding time in an exponential application algebraically (using logs)
(ii) Various Exponential Applications Solved Algebraically
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(17) For our midyear review in preparation for our performance assessment in the end of the unit on exponential functions, check the following PowerPoint presentation comparing linear, quadratic and exponential functions.
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(16) Solving exponential applications numerically and graphically (without logarithms, although some of the examples include the algebraic solution using logs side- by- side with the graphical or numerical solution).
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(15) If you need help using the graphing calculator to solve various equations (focus on exponential equations), you may want to see some worked- out examples below.
Graphing Calculator Tutorial Through Practice Examples on Solving Equations Graphically
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(14) If you need extra help on graphing simple exponential growth and exponential decay functions, here is a link to some examples that are explained step- by- step. Please note that on this website there is no graphing template, and they also show you a table of values to guide you in case that you needed that, but in class you are expected to learn how to use the graphing template for graphing exponential functions.
Intro Graphing Exponential Growth and Decay Functions
NOTE: The two links on the last page for additional examples and the solution key to the practice problems are not working right now. Sorry for any inconvenience. You may use a graphing calculator or ask your teacher if you need to check the graphs that you made by hand.
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(13) Here is a list of resources on working with complex numbers from the definition of the imaginary unit i to powers of i, operations with complex numbers (addition, subtraction, multiplication and division), plotting, finding the absolute value of a complex number, and solving quadratic equations involving complex numbers (using the Quadratic Formula or reverse PEMDAS/ finding square roots):
Intro Complex Numbers from PurpleMath
Intro Complex Numbers from Khan Academy
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(12) HONORS ALGEBRA II STUDENTS - Watch the YouTube video below for using the formulas for sum and difference of cubes for factoring polynomials.
Example of sum/ difference of cubes for factoring polynomials
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(11) HONORS ALGEBRA II STUDENTS - Watch the YouTube video (see link below) for finding all the zeros of a polynomial function using the Rational Zeros Theorem, along with repeated synthetic division, and other factoring strategies.
NOTE: I would highly recommend that you try to shorten the list of possible rational zeros by looking at the graph from your graphing calculator instead of using a trial/ error strategy that is presented in the video. I would also recommend that you use the Box method for factoring the quadratic expression that shows up in the second part of the solution below, rather than reverse FOIL, since the leading coefficient is not 1, and the process of guessing/ checking the two factors may take too much time.
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(10) If you need extra help for factoring polynomials (in particular quadratics learned in Algebra I) using the Box Method or finding GCFs, check some of the videos from YouTube below. If you still have trouble after that, please stay after school or during the X- block for extra help. You may have to copy the following link(s) in a different window, to be able to see the videos.
1. Box used for factoring a quadratic expression with leading coefficient 1 (review from Algebra I)
2. Box used for factoring a quadratic expression with leading coefficient different than 1
4. Finding GCFs for factoring polynomials
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(9) If you need extra help graphing a polynomial function in root form with single, double and triple roots, check out the videos below. These videos explain step- by- step how to sketch the graph just like we did in class. For the complete discussion of the graph, you may use the polynomial graphing template that we used in class.
1. Graph polynomial functions in root form (Note: For these examples, the author refers to "factored form", but he really talks about the "root form" like we discussed in class. The first example presented is a graph of a cubic polynomial with single roots. The second example includes a single root, a double root, and a triple root)
2.Graph a polynomial in root form with single, double and triple roots
3.More examples of graphing polynomial functions in root form with single and multiple roots
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(8) If you need extra help using synthetic division for polynomials, check out the extra practice problems worked- out (click the link below), then try some of the practice problems by yourself (see the bottom of that page for the link to the practice problems).
Extra Practice - Synthetic Division of Polynomials
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(7) If you need extra help on polynomial long division, check out the extra practice problems worked- out (click the link below) and try some of the practice problems after that (see the bottom of that page for a link to the practice problems).
Extra practice for polynomial long division
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(6) If you need extra help on adding, subtracting, and multiplying polynomials (including squaring binomials), check out the worked- out examples (click corresponding links below) and try some of the practice problems by yourself. The practice problems have the answers and the complete solutions available as well.
Extra Practice Adding and Subtracting Polynomials
Extra Practice Multiplying Polynomials
Extra Practice Squaring Binomials
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(5) If you need extra practice on graphing quadratic functions in root form, you may want to check out the following examples:
Extra Practice Graphing Quadratic Functions in Root (Intercept) Form
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(4) If you need extra practice on graphing quadratic functions in vertex form step- by- step, you may want to check out the following examples
Extra Practice Graphing Quadratics in Vertex Form
NOTE: You may skip finding the x-intercepts for now, and use different mirror points like we did in class. We will go over strategies for solving quadratic equations in the next unit. Stay tuned!
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(3) If you need extra help to graph a quadratic function in standard form, here is a quick example that follows almost the same steps that we used in our graphic organizer for graphing quadratic functions in standard form:
Extra Practice Graphing Quadratic Function in Standard Form
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(2) If you don't have a graphing calculator, you may use the following online graphing tool:
Graphing Calculator Tool (use this to graph a function if you already have its rule)
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(1) If you get stuck writing the closed rule for a quadratic function and you do not have a graphing calculator, you may want to use the following regression calculators online, which are quite simple to use:
1. Quadratic Regression Calculator (use this for quadratic patterns)
2. Regression Calculator for Linear, Exponential, Logarithmic, Power Rules (use the answer for Exponential Regression for the exponential rule or Power Regression for the rational rule)
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