Pre-Calculus U-46 Curriculum Scope and Sequence

Transcription

1 Pre-Calculus U-46 Curriculum Scope and Sequence Pre-Calculus 2018 Repting Strand Instructional Focus Common Ce Standards Sem. Compose and transfm functions F-BF-3(+), F-BF-1c Functions Produce inverse functions F-BF- 4 1 Graph and interpret rational functions F-IF-7 Exponential and Logarithmic Functions Graph and interpret exponential and logarithmic functions Use inverse relationships s F-IF-7, F-BF-3(+) F-BF-5 1 Series and Conics Use finite and infinite fmulas s A-SSE-4 (+) 1 Derive equations ellipses and hyperbolas G-GPE-2, G-GPE-3 1 Use unit circles and inverse trigonometric functions F-TF-3, F-TF-4, F-TF-6, F- TF-7 Trigonometry Graph and transfm trigonometric functions F-TF-4, F-BF-3(+), F-IF-7e 2 Codinate Systems Prove and use trigonometric functions Represent and calculate with vects Represent and calculate complex numbers G-SRT-9, G-SRT.10, G-SRT.11, F-TF-9, F-TF-8 N-VM-1, N-VM-2, N-VM-3, N-VM-4, N-VM-5 N-CN-4, N-CN-3, N-CN-5, N-CN-6 2 Limits Find limits and continuity Calculus Prep 2 1

2 Functions Instructional Focus: Compose and transfm functions Identify and Find Transfmations (F.BF.3) Compose Functions (F.BF.1c) Identify effect on a me than two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and me than two transfmations, find values constants and coefficients Given a partial graph, complete graph f both even and odd functions Evaluate composition 2 functions in context a situation Identify effect on a two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and two transfmations, find values constants and coefficients Recognize even and odd functions from graphs and equations Evaluate composition 2 functions Identify effect on a a single transfmation: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and a single transfmation, find value constant coefficient Recognize even and odd functions from graphs Evaluate a function f a given value and use that result to evaluate a second function F.BF.3 (+) Identify effect on graph replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) f specific values k (both positive and negative); find value k given graphs. Experiment with cases and illustrate an explanation effects on graph using technology. Include recognizing even and odd functions from ir graphs and algebraic expressions f m. F.BF.1c Compose F example, if T(y) is temperature in atmosphere as a function height, and h(t) is height a wear balloon as a function time, n T(h(t)) is temperature at location wear balloon as a function time. 2

3 Functions Instructional Focus: Produce inverse functions Produce inverse functions (F.BF.4) Compose functions to verify if one function is inverse anor function Read values an inverse function from a graph and table Produce an invertible function from a noninvertible function by restricting domain so that function is one-toone Compose functions to verify if one function is inverse anor function Read values an inverse function from a graph and table Identify a domain that that will produce an invertible function from a noninvertible function Given a simple function, find its inverse Read values an inverse function from a graph table Identify if a function is invertible from a graph F.BF.4 Find inverse b. (+) Verify by composition that one function is inverse anor. c. (+) Read values an inverse function from a graph a table, given that function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting domain. 3

4 Functions Instructional Focus: Graph and interpret rational functions Identify key features graphs (F.IF.7) The concentration C (in mg/dl), a certain prescription drug in a person's bloodstream is determined using rational function: where t is time (in hours) after taking prescription drug What is equation hizontal asymptote f graph function? What does this value (and fact that it is an asymptote) represent in context this? thinking beyond standard, including tasks that may involve Graph rational functions, given model, and interpret all related key features a graph in context a real wld situation. end behavi Graph rational functions, given model, and identify all related key features a graph. end behavi Given graphs rational, exponential, logarithmic and trigonometric functions, and identify all related key features a graph. end behavi F.IF.7 Graph functions expressed symbolically and show key features graph, by hand in simple cases and using technology f me complicated cases. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factizations are available, and showing end behavi. 4

5 Exponential and Logarithmic Functions Instructional Focus: Graph and interpret exponential and logarithmic functions Identify and Find Transfmations (F.BF.3) Identify key features graphs (F.IF.7) Identify effect on a me than two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and me than two transfmations, find values constants and coefficients Graph exponential and logarithmic functions, and interpret all related key features a graph in context a real wld situation. end behavi Identify effect on a two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and two transfmations, find values constants and coefficients Graph exponential and logarithmic functions, and identify all related key features a graph. end behavi Identify effect on a a single transfmation: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and a single transfmation, find value constant coefficient Given graphs exponential and logarithmic functions, and identify all related key features a graph. end behavi F.BF.3 (+) Identify effect on graph replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) f specific values k (both positive and negative); find value k given graphs. Experiment with cases and illustrate an explanation effects on graph using technology. Include recognizing even and odd functions from ir graphs and algebraic expressions f m. F.IF.7 Graph functions expressed symbolically and show key features graph, by hand in simple cases and using technology f me complicated cases. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factizations are available, and showing end behavi. e. (+) Graph exponential and logarithmic functions, showing intercepts and end behavi, and trigonometric functions, showing period, midline, and amplitude. 5

6 Exponential and Logarithmic Functions Instructional Focus: Use inverse relationships exponential and logarithmic s Exponential and Logarithmic inverses (F.BF.5) Recognize that exponential and logarithmic functions are inverses each or and use se functions to solve real-wld s. Recognize that exponential and logarithmic functions are inverses each or and use se functions to solve logarithmic and exponential equations. Recognize that exponential and logarithmic functions are inverses each or and convert from one fm into or. F.BF.5 (+) Understand inverse relationship between exponents and logarithms and use this relationship s involving logarithms and exponents. 6

7 Series and Conics Instructional Focus: Use finite and infinite fmulas s Finite and infinite fmulas (A.SSE.4) Use finite and infinite fmulas f geometric series real-wld s Use finite and infinite fmulas f geometric series to find: sum first term last term rate Find sum, using finite and infinite fmulas, f geometric series A.SSE.4 (edited) Use finite and infinite fmulas f geometric series s. F example, calculate mtgage payments. 7

8 Series and Conics Instructional Focus: Derive equation ellipses and hyperbolas Conics (G.GPE2, G.GPE.3) Write equation a parabola given its focus and directrix. Write standard equation an ellipse hyperbola given graph, foci, general fm equation. Identify center, vertices, and foci given equation an ellipse hyperbola Identify equation a parabola given its focus and directrix. Write standard equation a hyperbola ellipse given graph Identify center and vertices an ellipse hyperbola given graph equation Identify focus and directix a parabola Identify if a given equation represents an ellipse hyperbola Identify center an ellipse hyperbola given graph equation a level 1 G.GPE.2 Derive equation a parabola given a focus and directrix. G.GPE.3 (+) Derive equations ellipses and hyperbolas given foci, using fact that sum difference distances from foci is constant. 8

9 Trigonometry Instructional Focus: Use unit circles and inverse trigonometric functions Use special triangles (F.TF.3) Use unit circles to find values (F.TF.4) Construct Inverse trigonometric functions (F.TF.6) Use inverse trigonometric functions (F.TF.7) Pythagean identity (F.TF.8) Use special triangles to determine values sine, cosine, tangent, secant, cosecant, and cotangent f 0, π/6, π/4 and π/3, π/2 and use unit circle to express values sine, cosine, tangent, secant, cosecant, and cotangent f π x, π+x, and 2π x in terms ir values f x, where x is any real number Use unit circle to express any angle, including negative angles and angles involving me than 1 rotation, in terms its standard position to find all six trigonometric Construct an invertible trigonometric function by restricting domain so that function is always increasing decreasing Use inverse functions to solve trigonometric equations with restricted and unrestricted domains and interpret solutions in context situation Prove Pythagean identity ssiinn 2 (θθ) + ccooss 2 (θθ) = 1 and use it to find ssiinn (θθ), ccooss (θθ), and ttaann (θθ) Use special triangles to determine values sine, cosine, tangent, secant, cosecant, and cotangent f 0, π/6, π/4, π/3 and π/2 Use unit circle to express any angle, between 0 and 2ππ, in terms its standard position to find ALL 6 trig Identify a domain that will allow construction inverse a trigonometric function, because function would be always increasing decreasing Use inverse functions to solve trigonometric equations with restricted and unrestricted domains Use Pythagean identity ssiinn 2 (θθ) + ccooss 2 (θθ) = 1 to find ssiinn (θθ), ccooss (θθ), and ttaann (θθ) Use special triangles to determine values sine, cosine and tangent f π/6, π/4 and π/3 Use unit circle to express any angle, between 0 and 2ππ, in terms its standard position to find sine, cosine, and tangent Given a ption a trigonometric graph, identify if that part graph is invertible Use inverse functions to solve trigonometric equations with restricted domains Use Pythagean identity ssiinn 2 (θθ) + ccooss 2 (θθ) = 1 to find ssiinn (θθ), ccooss (θθ), ttaann (θθ) Pre-Calculus 2018 F.TF.3 (+) Use special triangles to determine geometrically values sine, cosine, tangent f π/3, π/4 and π/6, and use unit circle to express values sine, cosine, and tangent f π x, π+x, and 2π x in terms ir values f x, where x is any real number. Functions F.TF.4 (+) Use unit circle to explain symmetry (odd and even) and periodicity trigonometric F.TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing always decreasing allows its inverse to be constructed. F.TF.7 (+) Use inverse functions trigonometric equations that arise in modeling contexts; evaluate solutions using technology, and interpret m in terms context. F.TF.8 Prove Pythagean identity ssiinn 2 (θθ) + ccooss 2 (θθ) = 1 and use it to find ssiinn (θθ), ccooss (θθ), ttaann (θθ) given ssiinn (θθ), ccooss (θθ), ttaann (θθ) and quadrant angle. 9

10 Trigonometry Instructional Focus: Graph and transfm trigonometric functions Symmetry and periodicity trigonometric functions (F.TF.4) Identify and Find Transfmations (F.BF.3) Identify key features graphs (F.IF.7) Use unit circle to explain symmetry (odd and even) six trigonometric Use periodicity unit circle to explain repeated cycle graphs all six trigonometric Identify effect on a me than two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and me than two transfmations, find values constants and coefficients Given a partial graph, complete graph f both even and odd functions Graph trigonometric functions, and interpret all related key features a graph in context a real wld situation. period midline amplitude Use unit circle to explain symmetry (odd and even) sine, cosine, and tangent Use periodicity unit circle to explain repeated cycle graphs sine, cosine, and tangent Identify effect on a two transfmations: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and two transfmations, find values constants and coefficients Recognize even and odd functions from graphs and equations Graph trigonometric functions, and identify all related key features a graph. period midline amplitude Use unit circle to explain symmetry (odd and even) sine and cosine Use periodicity unit circle to explain repeated cycle graphs sine and cosine Identify effect on a a single transfmation: f(x) + k, k f(x), f(kx), f(x + k) f specific positive and negative values k Given graph a function and a single transfmation, find value constant coefficient Recognize even and odd functions from graphs Given graphs and trigonometric functions, and identify all related key features a graph. period midline amplitude Graphing F.TF.4 (+) Use unit circle to explain symmetry (odd and even) and periodicity trigonometric F.BF.3 (+) Identify effect on graph replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) f specific values k (both positive and negative); find value k given graphs. Experiment with cases and illustrate an explanation effects on graph using technology. Include recognizing even and odd functions from ir graphs and algebraic expressions f m. F.IF.7 Graph functions expressed symbolically and show key features graph, by hand in simple cases and using technology f me complicated cases. e. (+) Graph exponential and logarithmic functions, showing intercepts and end behavi, and trigonometric functions, showing period, midline, and amplitude. 10

11 Trigonometry Instructional Focus: Prove and use trigonometric functions Prove and use fmulas (F.TF.9) Derive area fmula (G.SRT.9) Law Sines and Cosines (G.SRT.10 and 11) Prove addition and subtraction fmulas f sine, cosine, and tangent and use addition and subtraction fmulas to solve identities Explain how to derive fmula: A = 1/2 ab sin(c) f area a triangle, and utilize it to find area a polygon composed multiple triangles Prove Law Sines and Law Cosines, and apply m to find unknown measurements in oblique triangles and interpret solutions in context realwld situations Prove addition and subtraction fmulas f sine, cosine, and tangent and use m numerical s Explain how to derive fmula: A = 1/2 ab sin(c) f area a triangle, and utilize it to find area a triangle Apply Law Sines and Law Cosines to find unknown measurements in oblique triangles Use addition, subtraction, and tangent fmulas numerical s Find area any triangle using fmula: A = 1/2 ab sin(c) Identify wher Law Sines Law Cosines should be applied to an oblique triangle to find unknown measurements, and if ambiguous case applies to triangle. F.TF.9 (+) Prove addition and subtraction fmulas f sine, cosine, and tangent and use m s. G.SRT.9 (+) Derive fmula A = 1/2 ab sin(c) f area a triangle by drawing an auxiliary line from a vertex perpendicular to opposite side. G.SRT.10 (+) Prove Laws Sines and Cosines and use m s. G.SRT.11 (+) Understand and apply Law Sines and Law Cosines to find unknown measurements in right and non-right triangles (e.g., surveying s, resultant fces). 11

12 Codinate Systems Instructional Focus: Represent and calculate with vects Represent vects (N.VM.1) Solve s with vects (N.VM.3) Operations with vects (N.VM.2, N.VM.4, N.VM.5) Use appropriate symbols f vects and ir magnitude, represent vect quantities by directed line segments, and find magnitude and direction vect quantities. Solve s involving velocity and or quantities by converting given direction and magnitude quantities into component vects, calculate resultant vect, and find resultant direction and magnitude angle between vects Find components a vect by subtracting codinates Add, subtract vects graphically and componentwise, and determine magnitude and direction Multiply a vect by a scalar and determine magnitude and direction Use appropriate symbols f vects and ir magnitude and represent vect quantities by directed line segments. Solve s involving velocity and or quantities by converting given direction and magnitude quantities into component vects, and calculate resultant vect Find components a vect by subtracting codinates Add, subtract vects graphically and componentwise Multiply a vect by a scalar Use appropriate symbols f vects and ir magnitude Solve s involving velocity and or quantities by converting given direction and magnitude quantities into component vects Find components a vect by subtracting codinates Add, subtract vects graphically componentwise Multiply a vect by a scalar N.VM.1 (+) Recognize vect quantities as having both magnitude and direction. Represent vect quantities by directed line segments, and use appropriate symbols f vects and ir magnitudes (e.g., v, v, v, v). N.VM.3 (+) Solve s involving velocity and or quantities that can be represented by vects. N.VM.2 (+) Find components a vect by subtracting codinates an initial point from codinates a terminal point. N.VM.4 (+) Add and subtract vects. a. Add vects end-to-end, component-wise, and by parallelogram rule. Understand that magnitude a sum two vects is typically not sum magnitudes. b. Given two vects in magnitude and direction fm, determine magnitude and direction ir sum. c. Understand vect subtraction v w as v + ( w), where w is additive inverse w, with same magnitude as w and pointing in opposite direction. Represent vect subtraction graphically by connecting tips in appropriate der, and perfm vect subtraction component-wise. N.VM.5 (+) Multiply a vect by a scalar. a. Represent scalar multiplication graphically by scaling vects and possibly reversing ir direction; perfm scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute magnitude a scalar multiple cv using cv = c v. Compute direction cv knowing that when c v 0, direction cv is eir along v (f c > 0) against v (f c < 0). 12

13 Codinate Systems Instructional Focus: Represent and calculate complex numbers Represent on complex plane (N.CN.4) Operations Vects in Polar Fm (N.CN.3, N.CN.5) Represent complex numbers on complex plane in rectangular and polar fm, and explain why rectangular and polar fms a given complex number represent same number Represent and compute addition and subtraction complex numbers geometrically on complex plane Represent and compute multiplication and division, in polar fm, complex numbers geometrically on complex plane Represent and compute power and roots complex numbers, in polar fm. Represent complex numbers on complex plane in rectangular and polar fm Represent and compute addition and subtraction complex numbers geometrically on complex plane Represent and compute multiplication and division, in polar fm, complex numbers geometrically on complex plane Represent complex numbers on complex plane in rectangular fm Represent and compute addition and subtraction complex numbers geometrically on complex plane Calculate distance and midpoint (N.CN.6) Calculate distance between numbers in complex plane as modulus difference, and calculate midpoint a segment in complex plane as average numbers at its endpoints Calculate difference between numbers in complex plane, and calculate midpoint a segment in complex plane as average numbers at its endpoints Calculate midpoint a segment in complex plane as average numbers at its endpoints N.CN.4 (+) Represent complex numbers on complex plane in rectangular and polar fm (including real and imaginary numbers), and explain why rectangular and polar fms a given complex number represent same number. N.CN.3 (+) Find conjugate a complex number; use conjugates to find moduli and quotients complex numbers. N.CN.5 (+) Represent addition, subtraction, multiplication, and conjugation complex numbers geometrically on complex plane; use properties this representation f computation. F example, ( i) 3 = 8 because ( i) has modulus 2 and argument 120. N.CN.6 (+) Calculate distance between numbers in complex plane as modulus difference, and midpoint a segment as average numbers at its endpoints. 13

14 Limits Instructional Focus: Find limits and continuity Find limits Determine continuity Find limits and one-sided limits graphically, numerically, and algebraically, using proper notation. Describe end behavi (as x approaches -) using limit notation Determine continuity functions graphically, numerically, and algebraically on its domain using three-part definition continuous Determine values f which a function is discontinuous, understand difference between removable and nonremovable discontinuities, and be able to redefine functions to make m continuous when possible. Find finite and infinite onesided limits, and describe asymptotes using limit notation. Find limits and one-sided limits graphically and numerically. Describe end behavi (as x approaches ) using limit notation). Determine continuity functions graphically and numerically on its domain using three-part definition continuous Determine values f which a function is discontinuous, and understand difference between removable and nonremovable discontinuities. Find finite and infinite onesided limits. Find limits and one-sided limits graphically and numerically Determine continuity functions graphically and numerically at a given value using three-part definition continuous Determine values f which a function is discontinuous. Determine wher a onesided limit is finite infinite. Find limits and one-sided limits graphically, numerically, and algebraically, using proper notation. Describe end behavi (as x approaches -) using limit notation. Determine continuity functions graphically, numerically, and algebraically on its domain using three-part definition continuous Determine values f which a function is discontinuous, understand difference between removable and nonremovable discontinuities, and be able to redefine functions to make m continuous when possible. Find finite and infinite one-sided limits, and describe asymptotes using limit notation. 14