


Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
1 
Classroom Resources: 
1 

1. Define the constant ein a variety of contexts.
Example: the total interest earned if a 100% annual rate is continuously compounded.
a. Explore the behavior of the function y=e^{x} and its applications.
b. Explore the behavior of ln(x), the logarithmic function with base e, and its applications. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
1 
Classroom Resources: 
1 

2. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
1 
Classroom Resources: 
1 

3. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
1 
Classroom Resources: 
1 

4. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
Example: (1+ âˆš3i)^{3}=8 because (1+ âˆš3i) has modulus 2 and argument 120^{o}. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

5. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
6 
Classroom Resources: 
6 

6. Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
14 
Classroom Resources: 
14 

7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.
a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

8. Explain that vector quantities have both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
Examples: v, v, v, v. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
3 
Learning Activities: 
3 

9. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

10. Solve problems involving velocity and other quantities that can be represented by vectors.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

11. Find the scalar (dot) product of two vectors as the sum of the products of corresponding components and explain its relationship to the cosine of the angle formed by two vectors.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
3 
Learning Activities: 
3 

12. Add and subtract vectors.
a. Add vectors endtoend, componentwise, and by the parallelogram rule, understanding that the magnitude of a sum of two vectors is not always the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c. Explain vector subtraction, v  w, as v + (w), where w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
7 
Learning Activities: 
3 
Classroom Resources: 
4 

13. Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise.
Example: c(v_{x}, v_{y}) = (cv_{x}, cv_{y})
b. Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cvÂ â‰ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

14. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

15. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems, extending to infinite geometric series.
Examples: calculate mortgage payments; determine the longterm level of medication if a patient takes 50 mg of a medication every 4 hours, while 70% of the medication is filtered out of the patient's blood. Unpacked Content



Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

16. Derive and apply the Remainder Theorem: 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). Unpacked Content


Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

17. Know and apply the Binomial Theorem for the expansion of (x + y)^{n} in powers of x and y for a positive integer, n, where x and y are any numbers. Unpacked Content


Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

18. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated cases, a computer algebra system. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

19. Add, subtract, multiply, and divide rational expressions.
a. Explain why rational expressions form a system analogous to the rational numbers, which is closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
2 
Classroom Resources: 
2 

20. Explain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a clearcut solution. Construct a viable argument to justify a solution method. Include equations that may involve linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, and trigonometric functions, and their inverses. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

21. Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

22. Represent a system of linear equations as a single matrix equation in a vector variable.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

23. Find the inverse of a matrix, if it exists, and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
12 
Classroom Resources: 
12 

24. Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features.
Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
8 
Classroom Resources: 
8 

25. 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. Extend from polynomial, exponential, logarithmic, and radical to rational and all trigonometric functions.
a. Find the difference quotient ^{f(x+Î"x)f(x)}/_{Î"x} of a function and use it to evaluate the average rate of change at a point.
b. Explore how the average rate of change of a function over an interval (presented symbolically or as a table) can be used to approximate the instantaneous rate of change at a point as the interval decreases.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
6 
Classroom Resources: 
6 

26. Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.
a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
1 
Classroom Resources: 
1 

27. Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.
Example: If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. Unpacked Content


Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

28. Find inverse functions.
a. Given that a function has an inverse, write an expression for the inverse of the function.
Example: Given f(x) = 2x^{3} or f(x) = (x + 1)/(x  1) for xÂ â‰ 1 find f^{1}(x).
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or a table, given that the function has an inverse.
d. Produce an invertible function from a noninvertible function by restricting the domain. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

29. Use the inverse relationship between exponents and logarithms to solve problems involving logarithms and exponents. Extend from logarithms with base 2 and 10 to a base of e. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

30. Identify the effect on the graph of replacing f(x) by f(x)+k, k Â· f(x), f(k Â· x), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Extend the analysis to include all trigonometric, rational, and general piecewisedefined functions with and without technology.
Example: Describe the sequence of transformations that will relate y=sin(x) and y=2sin(3x). Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
2 
Classroom Resources: 
2 

31. Graph conic sections from seconddegree equations, extending from circles and parabolas to ellipses and hyperbolas, using technology to discover patterns.
a. Graph conic sections given their standard form.
Example: The graph of ^{x2}/_{9} + ^{(y3)2}/_{4}=1 will be an ellipse centered at (0,3) with major axis 3 and minor axis 2, while the graphÂ ofÂ ^{x2}/_{9}Â +Â ^{(y3)2}/_{4}=1 will be a hyperbola centered at (0,3) with asymptotes with slope Â±3/2.
b. Identify the conic section that will be formed, given its equation in general form.
Example: 5y^{2 } 25x^{2}=25 will be a hyperbola.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

32. Solve applicationbased problems involving parametric and polar equations.
a. Graph parametric and polar equations.
b. Convert parametric and polar equations to rectangular form. Unpacked Content


Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

33. Use special triangles to determine geometrically the values of sine, cosine, and tangent for Ï€/3, Ï€/4, and Ï€/6, and use the unit circle to express the values of sine, cosine, and tangent for Ï€  x, Ï€ + x, and 2Ï€  x in terms of their values for x, where x is any real number. Unpacked Content

Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

34. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

35. Demonstrate that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
0 

36. Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
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Mathematics (2019) 
Grade(s): 9  12 
Precalculus 
All Resources: 
3 
Classroom Resources: 
3 

37. Use trigonometric identities to solve problems.
a. Use the Pythagorean identity sin^{2} (θ) + cos^{2}(θ) = 1 to derive the other forms of the identity.
Example: 1 + cot^{2} (θ) = csc^{2} (θ)
b. Use the angle sum formulas for sine, cosine, and tangent to derive the double angle formulas.
c. Use the Pythagorean and double angle identities to prove other simple identities. Unpacked Content
