Functions

Standard	Essential Question		Bloom’s Taxonomy Activities	Vocabulary	Pacing
F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y =f(x).	How are domain and range related to each other? How does f relate to the graph of y=f(x)? Why are tables important when considering the input and outputs of equations?		-Support the concept that the domain of a number matches to exactly one element of the range. - Develop a linear function demonstrating the relationship between domain and range.	-Set -Domain -Range -Element -Function -Input -Output	2 days
F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.	How does understanding function notation assist in creating graphs by hand?		-Convert equations in y= form to function notation. -Construct models of linear and exponential models on paper and on a graphing calculator.	-Function Notation	2 days
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1), f(n+1) = f(n) + f(n-1) for n≥1.	How could sequences be defined in function notation?		-Create a recursive sequence or formula. - Examine the relationship between functions and sequences.	-Sequence -Subset -Recursive	2 days
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and table 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; relative maximums and minimums; symmetries; end behaviors; and periodicity.*	How do relative maximums and relative minimums relate to where a graph is increasing or decreasing? What relationships do intercepts have with the axis?		-Create two equations and graph them on paper. -Contrast the key features of the equations and their respective graphs.	-X-Intercept -Y-Intercept -Interval -Increasing -Decreasing -Positive -Negative -Relative Maximum -Relative Minimum -Symmetry -End Behavior	3 days
F.IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*	Why is it important to reread the question before providing a final answer? How do questions pertaining to time, people, and distance affect the domain of an answer?		-Examine the responses to questions involving domain and word problems to discover the quantitative relationship between what the question is asking and the response.	-Domain -Function	3 days
F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*	How do points on a linear graph assist in determining rate of change?		-Approximate the rate of change from a given graph. -Estimate the rate of change for a quadratic function for each interval.	-Average -Rate of Change -Interval	3 days
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*	What is the relationship between the parts of a function and its graph?		- Examine a function expressed symbolically to create its graph. -Judge a given function to determine if it is best to complete manually or through technology.	-Function -Approximation	3 days
F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minim	How can the qualities of a function expressed symbolically assist in creating its graph?		-Compare and contrast various functions to determine the attributes of their corresponding graphs.	-Linear -Quadratic -Intercept -Maxima -Minima	3 days
F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.	What is the relationship between a graph to an integer power and a graph to a fractional power?		-Create a poster board with a sample of each of the following graphs: square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graphs may be hand-drawn or created on the computer using a program such as Microsoft Mathematics	-Square root -Cube root -Piece-wise function -Step function -Absolute value function	3 days
F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.	How can technology support the understanding of polynomial equations?		-Create a PowerPoint slide based on a given equation; include two graphics on the slide, 1) a hand-drawn graph of the equation and 2) an electronic representation of the polynomial	-Polynomial functions -Factorizations	3 days
F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.	What is the relationship, if any, between the qualities of logarithmic graphs and exponential graphs?		-Create an equation for exponential and logarithmic graphs and provide information regarding intercepts and end behavior	-Exponential function - Logarithmic function -End behavior	3 days
F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of functions.	What properties can be used to express functions in equivalent forms? Why is it beneficial to express functions in equivalent forms?		-Create a visual highlighting the different ways a function can be expressed	-Function -Equivalent forms	3 days
F.IF.8a 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.	In what instances would completing the square be beneficial in simplifying functions?		-Distinguish the qualities of a graph based on the result of a simplified function	-Factoring -Completing the square -Quadratic function -Extreme values -Symmetry	3 days
F.IF.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y=(0.97)t, y=(1.01)12t, y=(1.2)t/10, and classify them as representing exponential growth or decay.	What are three examples in which exponential functions are used in the real world?		-Construct a real-world example of the use of an exponential equations, share your problem with your class, and display your work	-Exponents -Exponential function	3 days
F.IF.9 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.	What qualities of a function expressed symbolically assist in determining the features of the graph?		-Design a visual representation comparing two self-created function in four representations: numerically in tables, graphically, verbally, and algebraically	-Properties of functions	3 days
Standard	Essential Question		Bloom’s Taxonomy Activities	Vocabulary	Pacing
F.BF.1 Write a function that describes a relationship between two quantities.*	What are some context clues in a word problem which assist in creating a function which depicts the relationship between the quantities provided?		-Analyze a word problem to create a function which accurately depicts the relationship between the two quantities.	-Relationship between functions -Sum -Difference -Increase -Divided	3 days
F.BF.1a Determine an explicit expression, a recursive process, or steps from a context.	What are the benefits of creating an explicit expression for a given sequence?		-Distinguish when writing an explicit expression is more valuable than determining a recursive process or equation.	-Explicit expression -Recursive process -Sequence	3 days
F.BF.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.	When is it necessary to combine functions in solving a problem?		-Create a word problem where combining functions algebraically is necessary.	-Standard function	3 days
F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*	What is the relationship between geometric and arithmetic sequences? In determining equations to a high term, which form is best?		-Given an arithmetic or geometric sequence, create two formulas: one recursive and one explicit.	-Arithmetic sequence -Geometric sequence	3 days
F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) +  k. k f(x), f(kx), and f(x+k) for the 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.	How does including an additional graph affect the graph of a function?		-Develop four scenarios in which a graph is increased or decreased by the constant, k	-Constant -Translation -Transformation	1 day
F.BF.4 Find the inverse functions.	Why is knowledge of inverse functions beneficial when considering purchasing a house?		-Choose a house you would be interested in purchasing and research financing options, then select which finance plan is best.	-Inverse function	1 day
F.BF.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x-1) for x≠1.	What is the relationship of domain and range in a function?		-Create a worksheet and answer key of no less than seven problems of simple functions for peers to complete the inverse	-Inverse of a function -Domain -Range	1 day
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Standard	Essential Question	Bloom’s Taxonomy Activities		Vocabulary	Pacing
F.LE.1 Distinguish between situations that can be modeled    with linear functions and with exponential functions.	What is the difference between the progression of points on a linear graph and the progression of points on an exponential graph?	-Given a situation, determine if the model created is linear or exponential based on the context clues in the equation.		-Linear graph -Exponential graph	3 days
F.LE.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.	How many points are needed to create an accurate equation for a linear function and an accurate equation for an exponential function? Are there exceptions?	-Given the graph of a linear function, defend algebraically or graphically that the function increases or decreases over equal intervals by equal quantities. -Given the graph of an exponential function, prove growth occurs over equal intervals.		-Interval -Equal factor -Equal difference	3 days
F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.	What context clues suggest that a quantity changes at a constant rate per interval?	-Analyze several word problems relating to constant rate of change to determine which qualities or context clues are consistent across the problems.		-Constant rate	3 days
F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.	What context clues suggest that a quantity changes at a constant factor per interval?	-Analyze several word problems relating to constant factors of change to determine which qualities or context clues are consistent across the problems.		-Constant factor	3 days
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).	What information is necessary to create a graphical or algebraic representation of a described function?	-Given various expressions of information pertaining to a linear or an exponential function, create a graphical or algebraic representation of the function.		-Input-output pairs	3 days
F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.	What is unique about an exponential function as compared to a linear or quadratic function?	-Analyze linear, quadratic, and exponential functions to determine unique qualities between the three types of functions.		-Exponential function -Linear function -Quadratic function	3 days
F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.	In what instances is 2+2 not equal to 4?	-Self-guide your understanding of logs using the graphing calculator; create your own problems and share with the class http://mathbits.com/mathbits/tisection/Algebra2/logarithms.htm		-Logarithm -Base -Exponent -Natural log	3 days
F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.	How does the manipulation of the components of a function relate to its corresponding graph?	-Given a function, manipulate various components such as slope or intercept to determine how the function changes under the new conditions.		-Parameters	1 day
Standard	Essential Question	Bloom’s Taxonomy Activities		Vocabulary	Pacing
F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.	What is the difference between radian measure and degree measure?	-Compare the radian measures of various unit circles and their corresponding central angles.		-Unit circle -Radian measure -Central angle	1 day
F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.	Why is the unit circle used to determine radian measures?	-Create a visual of a unit circle with positive and negative radian measures		-Unit circle -Coordinate plane -Trigonometric	1 day
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F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.*	What consistencies exist when analyzing trigonometric functions?	-Investigate various online trigonometry resources at http://www.homeschoolmath.net/online/trigonometry.php to experiment with amplitude, frequency, and midline of trigonometry functions		-Sine -Cosine -Tangent -Amplitude -Frequency -Midline -Periodic	1 day
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F.TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) =1 and use it to find sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.	How is knowledge of the Pythagorean Theorem useful in determining trigonometric identities?	-Use the Pythagorean theorem to prove sin2(θ) + cos2(θ) =1. Use the identity to create at least three examples. Resource: http://cullinan.home.mchsi.com/Lesson_8.pdf		-Pythagorean identity	1 day