A sixteen years old learner is expected to be able to:

 

 

1. Extend polynomial identities to the complex numbers. For example, rewrite as (x + 2i)(x – 2i).

 

 

 

2. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials

 

 

 

 

3. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

 

 

 

 

 

4. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

 

 

 

 

 

5. Add and subtract vectors.

• Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
• Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
• Understand 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 component-wise.

 

 

 

 

 

 

6. Multiply a vector by a scalar. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

 

 

 

 

 

7. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

 

 

 

8. 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. For example, calculate mortgage payments.

 

 

 

9. Know 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).

 

 

 

10. Identify zeros of polynomials when suitable factorizations are available.

 

 

 

11. 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 examples, a computer algebra system.

 

 

 

12. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

 

 

 

13. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases

• Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
• Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

 

 

 

14. Write a function that describes a relationship between two quantities.

• 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.
• Compose functions. For 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.

 

 

 

 

15. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

 

 

 

16. 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

 

 

 

 

 

17. Find inverse functions. Verify by composition that one function is the inverse of another.

 

 

 

 

18. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

 

 

 

19. Distinguish between situations that can be modeled with linear functions and with exponential functions.

• Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
• Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
• 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.

 

 

 

20. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

 

 

 

21. 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.

 

 

 

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

 

 

 

 

23. Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

 

 

 

24. 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.

• Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems

 

 

 

 

25. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

 

 

 

 

26. Prove the Laws of Sines and Cosines and use them to solve problems.

 

 

 

 

27. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

 

 

 

28. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

 

 

 

29. Derive the equation of a parabola given a focus and directrix.

 

 

 

30. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.