MAEHL, AMANDA

ALL WORK WILL BE POSTED ON GOOGLE CLASSROOM.

Overview 
This course is intended to build upon student work with linear, quadratic, and exponential functions. Students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. Through the Mathematical Practice Standards students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Syllabus

Unit 1

• Simplify radicals when given perfect squares and non-perfect squares.  

• Solve multistep equations (adding, subtracting, multiplying, dividing, distributing).  

• Factor basic polynomials.  

• Use the graphing calculator to determine key values, and describe graphs of quadratics and linear functions.  

• Calculate and interpret average rate of change of a function.  

• Graph linear equations using concepts of slope and intercepts. 

• Solve systems of linear equations using substitution, elimination and graphing.

Unit 2

• Complex Numbers 

  • Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real. 

  • Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 

• Solving Quadratics 

  • Solve Quadratic Equations for real and complex solutions applying a variety of methods including square roots, completing the square, the quadratic formula, factoring and the zero product property. 

  • Write complex solutions as a ± bi for real numbers a and b. 

  • Solve quadratic equations by inspection (e.g., for

    ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as
    for real numbers
    and
    . 

  • Use a graphing calculator to check solutions of quadratic equations. 

  • Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions. 

• Graphing Quadratics (Focus on extending learning from Algebra 1) 

  • Identify key characteristics of quadratic graphs including the axis of symmetry, vertex, maximum/minimum values, xintercepts, y-intercepts, domain, range and intervals of increasing and decreasing.

  • Use features of the graphing calculator, such as the table, to understand characteristics of quadratic functions.  

  • Graph Quadratic Equations from standard form, and vertex form Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. 

  • Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

  • For quadratic functions, identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.  

  • For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

  • Recognize and identify even and odd quadratic functions from their graphs and algebraic expressions for them. 

  • Relate the domain of a quadratic function to its graph and, where applicable, to the quantitative relationship it describes. 

  • Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

  • Estimate the rate of change over a specific interval. 

  • Use the quadratic formula to solve real life application problems.

• Systems of Equations 

  • Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line

    and the circle
    .

  • Solve a systems with 3 unkown variables. 

  • Explain why the x-coordinates of the points where graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equations f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

Unit 3

• Definitions of Polynomials  

  • Recognize characteristics of a polynomial expression and use proper vocabulary to classify polynomial expressions.  

  • Express polynomials in standard form.  

  • Perform arithmetic operations on polynomial expressions: add, subtract & multiply.

• Solving Polynomials  

  • Understand the relationship between zeros and factors of polynomials.  

  • Factor and solve polynomials using the zero product property for all solutions (real and complex) above degree 2, including:  o Quartic Polynomials o Greatest Common Factor o Embedded Difference of Squares o Using Sums/Differences of Cubes o By Grouping (4 terms)

• Division of Polynomials  

  • Use the Remainder Theorem to evaluate polynomials.  (For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x))  

  • Use synthetic division to divide polynomials given one possible factor or zero.  

  • Use synthetic division to solve polynomial equations given one factor or zero.  

  • Use synthetic division to solve polynomial equations using the graphing calculator to determine a zero.

• Graphing Polynomials  

  • Create a basic graph of a polynomial: Identify zeros, multiplicity, and show end behavior.  

  • Identify zeros of polynomials by factoring by grouping or using synthetic division (given one zero or factor), and use the zeros to construct a rough graph of the function defined by the polynomial. 

  • Identify zeros of polynomials by using synthetic division and the graphing calculator to determine a zero, and use the zeros to construct a rough graph of the function defined by the polynomial. 

  • Create an equation of a polynomial given zeros or from a graph.  

  • Recognize and identify even and odd functions from their graphs and algebraic expressions for them.

  • Interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; end behavior; and multiplicities.  Determine relative maximums and minimums.

  • Estimate the rate of change from a graph, table, or polynomial function over a given interval.

Unit 4

Seeing Structure in Expressions: Equivalent Forms  

• Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.    For example, we define

  to be the cube root of
  because we want
  to hold, so
  must equal 5. 

• Apply the properties of exponents to simplify expressions including positive and negative integers and fractional exponents.  

• Rewrite radical expressions using rational exponents and vice versa.  

• Rewrite expressions involving radicals and rational exponents using the properties of exponents.  

• Simplify radicals, including algebraic radicals (e.g.

  , simplify
  ). 

• Evaluate nth roots of real numbers using both radical notation and rational exponent notation.  

• Write equivalent expressions for exponential functions using the properties of exponents. (For example, rewrite exponential function f(x)=3x∙23x in the form f(x) = a(bx).  

• Simplify radical expressions using absolute value symbols when appropriate.

  • Use the structure of an expression to identify ways to rewrite it. For example, see

    
    as 
    , thus recognizing it as a difference of squares that can be factored as
    .

  Solving all types of Exponential Equations  

  • Solve equations with radicals and rational exponents.  

  • Solve equations with extraneous solutions.  

  • Solve exponential equations with like bases.  

  • Solve exponential equations with unlike bases using an equivalent form to rewrite with like bases

  • Find inverse functions. 

    • Solve for the inverse of radical functions. 

    • Solve an equation of the form

      for a simple function f that has an inverse and write an expression for the inverse. For example,

       

      Graphing Exponential, and Radical Functions  

      • Find and interpret the domain of radical functions.  

      • Graph radical functions using a graphing calculator and table function.  

      • Identify domain and range of exponential and radical functions.  

      • Identify increasing and decreasing behavior of exponential and radical functions.  

      • Identify and write end behavior of exponential and radical functions in various notations.  

      • Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. (Compare to finding the average rate of change for a linear function.)  

      • State the domain and range of exponential function  

      • Sketch the graph of exponential function showing intercepts, key points, asymptotes, and end behavior.  

      • Identify the effect on the graph of exponentials by replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. 

      • Experiment with cases and illustrate an explanation of the effects on the graph using technology. 

       

      Applications  

      • Apply problem solving to applicational problems including exponential functions.  

      • Solve real-world problems including exponential growth and decay.  

      • Use graphing technology to relate key characteristics of an exponential graph to real-world problems.

Unit 5

Logarithms 

• Understand the inverse relationship between exponents and logarithms. 

• For exponential models, express as a logarithm the solution to

  where
  ,
  , and
  are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 

• Write exponential functions in log form and logarithmic functions in exponential form.

• Expand and condense logarithmic expressions. 

• Solve logarithmic equations with base 2, 10 or e.

• Show solutions to logarithmic equations in either exact form or approximate (rounded decimals) when prompted to do so. 

• Check for extraneous solutions when solving logarithmic equations.

Graphing Logarithms  

• State the domain and range of logarithmic functions. 

• Sketch the graph of a logarithmic function, showing intercepts, key points, asymptotes, and end behavior by using technology. 

• Identify the effect on the graph of exponentials and logarithms replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Applications of Logarithms

• Apply properties of exponentials and logarithms to solve real world application problems.

Find inverse functions of Logarithms 

• Know how to find the inverse for logarithmic function and use this to solve exponential functions.

• Solve an equation of the form

  for a simple function f that has an inverse and write an expression for the inverse. For example,

Unit 6

• Rational Functions Rewrite simple rational expressions in different forms using inspection.  

• Multiply and divide rational expressions and identify extraneous solutions.

• Add and subtract rational expressions and identify extraneous solutions.

• Find the inverse of a rational function and identify the domain and range for the function and its inverse. 

• Find inverse functions. Solve an equation of the form

  for a simple function f that has an inverse and write an expression for the inverse. For example,  
  for
  . 

• Solve simple rational equations in one variable and use them to solve problems, justify each step in the process and the solution. 

• Show how extraneous solutions may arise when solving a rational equation. 

  • Use the graphing calculator to graph rational functions using the table feature as well as identify key characteristics.

  • Identify increasing and decreasing intervals as well as write the end behavior in limit notation 

  • Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Unit 7

Sequences

• Write the explicit and recursive rules for arithmetic and geometric sequences. 

• Examine sequences to identify as linear or exponential. 

  • Students are encouraged to use the graphing calculator, if necessary, to graph the functions.

• Examine arithmetic and geometric sequences to construct linear and exponential functions and graphs of such. 

• Write the explicit rule for a sequence given recursively and vice versa. 

• Write the recursive and explicit rules if possible for non-arithmetic or non-geometric sequences including squares, cubes, and Fibonacci. 

  • Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 

Series 

• Write a series with summation notation. 

• Evaluate the sum of a series in summation notation. Calculate the sum of finite geometric series. 

  • Use the formula for the sum of a finite geometric series to solve problems. 

  • Estimate the rate of change from an explicit or recursive rule, graph or table.

    • Solve real world applications using sequence and series formulas

  Applications of Sequences and Series

Unit 8

Summarize, represent, and interpret data on a single count or measurement variable. 

• Represent data with plots on the real number line (dot plots, histograms, and box plots). 

• Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets 

• Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 

• Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. 

• Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Summarize, represent, and interpret data on two categorical and quantitative variables 

• Summarize categorical data for two categories in two-way frequency tables. 

• Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). 

• Recognize possible associations and trends in the data.

• Represent data on two quantitative variables on a scatter plot and describe how the variables are related. 

  • Fit a function to the data (including with the use of technology). 

  • Use functions fitted to data to solve problems in the context of the data. 

  • Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. 

  • Informally assess the fit of a function by plotting and analyzing residuals, including with the use of technology.