Skill Descriptor: Analyze arithmetic and geometric sequences and series.

Global Competencies: CM, CTPS

Achievement Indicators:

• Define arithmetic and geometric sequences and arithmetic and geometric series
• Justify if given sequences are arithmetic or geometric (or neither)
• Determine the general term of arithmetic sequences or geometric sequences using a rule
• Determine the first term, common difference, number of terms, and general term for arithmetic sequences and arithmetic series
• Determine the first term, common ratio, number of terms, and general term for geometric sequences and geometric series
• Define the sum of number of terms of arithmetic or geometric series using sigma notation
• Apply a rule for determining the sum of a given number of terms of an arithmetic or geometric series
• Determine whether geometric series are convergent or divergent
• Generalize and apply a rule for determining the sum of an infinite geometric series
• Solve contextual problems that involve arithmetic or geometric sequences or series

Skill Descriptor: Factor polynomial expressions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Identify when specialized factoring techniques can be used to solve problems involving polynomials
• Factor polynomials using sum and difference of cubes, factor by grouping (including to get a difference of squares), and factoring quartics as trinomials
• Divide a polynomial expression by a binomial expression of the form 𝑥 – 𝑎 where 𝑎 is an integer by using long division and synthetic division
• Explain and apply the remainder and factor theorems

Skill Descriptor: Graph and analyze polynomial functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Justify if functions are polynomial functions given their equations
• Explain the effect of the constant term on the graph of the function
• Explain the end behaviour of the graph based on the leading term of the function
• Explain the relationship between the factors, zeros, roots, and 𝑥 -intercepts of a function
• Explain how the multiplicity of a zero of a function affects the graph
• Sketch the graph of a polynomial function
• Model contextual situations to solve problems
• Determine a possible equation of a function from a given table or graph

Skill Descriptor: Graph and analyze reciprocal functions.

Global Competencies: CTPS

Achievement Indicators:

• Compare the characteristics of a function and its reciprocal including invariant points
• Identify the domains of reciprocal functions
• Graph the reciprocal given the graph of the function

Skill Descriptor: Graph and analyze rational functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Graph a rational function
• Determine if a non-permissible value results in a vertical asymptote or a removable discontinuity on the graph
• Explain the behaviour of the graph near non-permissible values (e.g., approaches positive or negative infinity; approaches a value)
• Match functions to their graphs
• Determine an approximate solution of a rational equation from a graph
• Describe the relationship between the roots of a rational equation and the 𝑥 -intercepts of the graph of the corresponding rational function
• Determine the equations of asymptotes

Skill Descriptor: Combine and compose functions $\begin{array}{l}f\left(f\right(x\left)\right)\\ f\left(g\right(x\left)\right)\\ g\left(f\right(x\left)\right)\\ f\circ g\left(x\right)\\ g\circ f\left(x\right)\\ f\circ f\left(x\right)\end{array}$ .

Global Competencies: CTPS

Achievement Indicators:

• Sketch the graph of a function that is the combination of two functions given their graphs
• Write the equation of a function that is the combination of two or more given functions
• Determine the domain and range of a function that is the combination of two functions
• Write a given function as the combination of two or more functions
• Determine the value of a composite function when evaluated at a point
• Determine the equation of a composite function given the equations of two functions
• Sketch the graph of a composite function given the equations of two functions
• Write a given function as a composition of two or more functions
• Write a function by combining and composing two or more functions
• Create compositions and inverses of all types of functions where appropriate
• Graph compositions of all types of functions
• Identify composition as an operation taking two functions as input and outputting the third function
• Apply definition of inverse functions as a property of operation of composition to verify two functions are inverses

Skill Descriptor: Analyze the fundamental counting principle.

Global Competencies: CM, CTPS

Achievement Indicators:

• Count the total number of possible choices that can be made using graphic organizers
• Explain the fundamental counting principle
• Apply the fundamental counting principle to solve problems

Skill Descriptor: Apply permutations to solve problems.

Global Competencies: CTPS

Achievement Indicators:

• Count the number of ways of ordering the elements of a finite set
• Determine the number of permutations for all elements of a finite set using factorial notation
• Determine the number of permutations of 𝑛 different elements taken 𝑟 at a time
• Solve an equation that involves permutation notation by finding the variable 𝑛 or 𝑟
• Analyze the effect on the total number of permutations when two or more elements of the set are identical

Skill Descriptor: Apply combinations to solve problems.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain the difference between a permutation and a combination
• Determine the number of combinations of 𝑛 different elements taken 𝑟 at a time
• Explain why 𝑛 must be greater than or equal to 𝑟 in permutation and combination notation
• Explain why 𝑛𝐶𝑟 = 𝑛𝐶𝑛−𝑟
• Solve an equation that involves combination notation by finding the variable 𝑛 or 𝑟

Skill Descriptor: Expand powers of a binomial.

Global Competencies: CM, CTPS

Achievement Indicators:

• Explain the patterns found in the expanded form of (𝑥 + 𝑦)ⁿ, 𝑛 ≤ 4, by multiplying 𝑛 factors of (𝑥 + 𝑦)
• Explain how to determine any row in Pascal’s triangle given a previous row
• Relate the coefficients of the terms in the expansion of (𝑥 + 𝑦)ⁿ to the (𝑛 + 1)^th row in Pascal’s triangle
• Explain how the coefficients of the terms in the expansion of (𝑥 + 𝑦)ⁿ are determined using combinations
• Expand (𝑥 +𝑦)ⁿ using the binomial theorem
• Determine a specific term in the expansion of (𝑥 + 𝑦)ⁿ

Skill Descriptor: Determine the limit of a function at a point.

Global Competencies: CTPS

Achievement Indicators:

• Demonstrate an understanding of limit notation
• Evaluate the limit of a function at a point graphically, analytically, and using tables of values
• Distinguish between the limit of a function as 𝑥 approaches 𝑎 and the value of the function at 𝑥 = 𝑎
• Evaluate a limit by substitution using the properties of limits
• Demonstrate an understanding of undefined limits
• Solve limits when substituting yields an indeterminate form

Skill Descriptor: Analyze one-sided limits.

Global Competencies: CTPS

Achievement Indicators:

• Demonstrate an understanding of the correct notation for one-sided limits
• Analyze one-sided limits as 𝑥 approaches a certain value
• Evaluate one-sided limits of non-continuous functions

Skill Descriptor: Analyze the continuity of a function.

Global Competencies: CTPS

Achievement Indicators:

• Demonstrate an understanding of the definition of continuity
• Determine if a function is continuous at a given point
• Determine if a function is continuous on a given interval
• Apply the Intermediate Value Theorem to continuous intervals of a function

Skill Descriptor: Analyze limits which involve infinity.

Global Competencies: CTPS

Achievement Indicators:

• Demonstrate an understanding of the properties of limits involving infinity
• Determine the horizontal asymptotes of a function using limits
• Evaluate limits of rational functions involving infinity
• Explore trigonometric graphs to show limits at infinity as an oscillating function

Skill Descriptor: Determine upper and lower bounds of sequences.

Global Competencies: CTPS

Achievement Indicators:

• Determine the bounds of increasing sequences
• Compare bounds of increasing, decreasing, constant, and damped oscillation sequences
• Determine the bounds of slopes of tangent lines