A: Quadratic Functions

A.1: expand and simplify quadratic expressions, solve quadratic equations, and relate the roots of a quadratic equation to the corresponding graph;

A.1.1: pose problems involving quadratic relations arising from real-world applications and represented by tables of values and graphs, and solve these and other such problems (e.g., “From the graph of the height of a ball versus time, can you tell me how high the ball was thrown and the time when it hit the ground?”)

Exponential Functions

A.1.3: factor quadratic expressions in one variable, including those for which a is not equal to 1 (e.g., 3x² + 13x - 10), differences of squares (e.g., 4x² - 25), and perfect square trinomials (e.g., 9x² + 24x + 16), by selecting and applying an appropriate strategy

Factoring Special Products
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

A.1.4: solve quadratic equations by selecting and applying a factoring strategy

Modeling the Factorization of x²+bx+c
Quadratics in Factored Form
Roots of a Quadratic

A.1.5: determine, through investigation, and describe the connection between the factors used in solving a quadratic equation and the x-intercepts of the graph of the corresponding quadratic relation

Roots of a Quadratic

A.1.6: explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numeric example; follow a demonstration of the algebraic development, with technology, such as computer algebra systems, or without technology [student reproduction of the development of the general case is not required]), and apply the formula to solve quadratic equations, using technology

Roots of a Quadratic

A.1.7: relate the real roots of a quadratic equation to the x-intercepts of the corresponding graph, and connect the number of real roots to the value of the discriminant (e.g., there are no real roots and no x-intercepts if b² – 4ac < 0)

Roots of a Quadratic

A.1.8: determine the real roots of a variety of quadratic equations (e.g., 100x² = 115x + 35), and describe the advantages and disadvantages of each strategy (i.e., graphing; factoring; using the quadratic formula)

Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic

A.2: demonstrate an understanding of functions, and make connections between the numeric, graphical, and algebraic representations of quadratic functions;

A.2.1: explain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, function machines, equations) and strategies (e.g., using the vertical-line test)

Addition and Subtraction of Functions
Arithmetic Sequences
Compound Interest
Linear Functions
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Roots of a Quadratic
Slope-Intercept Form of a Line
Translating and Scaling Functions
Zap It! Game

A.2.5: determine, through investigation using technology, the roles of a, h, and k in quadratic functions of the form f(x) = a(x – h)² + k, and describe these roles in terms of transformations on the graph of f(x) = x² (i.e., translations; reflections in the x-axis; vertical stretches and compressions to and from the x-axis)

Exponential Functions
Quadratics in Vertex Form
Translating and Scaling Functions
Translations
Zap It! Game

A.2.6: sketch graphs of g(x) = a(x – h)² + k by applying one or more transformations to the graph of f(x) = x²

Exponential Functions
Quadratics in Vertex Form
Translating and Scaling Functions
Translations
Zap It! Game

A.2.7: express the equation of a quadratic function in the standard form f (x) = ax² + bx + c, given the vertex form f (x) = a(x – h)² + k, and verify, using graphing technology, that these forms are equivalent representations

Quadratics in Vertex Form

A.2.8: express the equation of a quadratic function in the vertex form f(x) = a(x – h)² + k, given the standard form f(x) = ax² + bx + c, by completing the square (e.g., using algebra tiles or diagrams; algebraically), including cases where b/a is a simple rational number (e.g., ½, 0.75), and verify, using graphing technology, that these forms are equivalent representations

Quadratics in Vertex Form

A.2.10: describe the information (e.g., maximum, intercepts) that can be obtained by inspecting the standard form f(x) = ax² + bx + c, the vertex form f(x) = a(x – h)² + k, and the factored form f(x) = a(x – r)(x – s) of a quadratic function

Graphs of Polynomial Functions
Polynomials and Linear Factors
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form

A.2.11: sketch the graph of a quadratic function whose equation is given in the standard form f(x) = ax² + bx + c by using a suitable strategy (e.g., completing the square and finding the vertex; factoring, if possible, to locate the x-intercepts), and identify the key features of the graph (e.g., the vertex, the x- and y-intercepts, the equation of the axis of symmetry, the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing)

Quadratics in Factored Form
Quadratics in Polynomial Form

A.3: solve problems involving quadratic functions, including problems arising from real-world applications.

A.3.3: solve problems arising from real-world applications, given the algebraic representation of a quadratic function (e.g., given the equation of a quadratic function representing the height of a ball over elapsed time, answer questions that involve the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement)

Addition and Subtraction of Functions
Quadratics in Polynomial Form

B: Exponential Functions

B.1: simplify and evaluate numerical expressions involving exponents, and make connections between the numeric, graphical, and algebraic representations of exponential functions;

B.1.4: determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f(x) = a to the x power (a > 0, a is not equal to 1), function machines]

Exponential Functions
Logarithmic Functions

B.1.5: determine, through investigation (e.g., by patterning with and without a calculator), the exponent rules for multiplying and dividing numeric expressions involving exponents [e.g., (½)³ x (½)²], and the exponent rule for simplifying numerical expressions involving a power of a power [e.g., (5³)²], and use the rules to simplify numerical expressions containing integer exponents [e.g., (2³)(2 to the 5th power) = 2 to the 8th power]

Dividing Exponential Expressions
Exponents and Power Rules
Multiplying Exponential Expressions

B.1.6: distinguish exponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations), within the same context when possible (e.g., simple interest and compound interest, population growth)

Exponential Functions

B.2: identify and represent exponential functions, and solve problems involving exponential functions, including problems arising from real-world applications;

B.2.2: identify exponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and explain any restrictions that the context places on the domain and range (e.g., ambient temperature limits the range for a cooling curve)

Exponential Growth and Decay

B.2.3: solve problems using given graphs or equations of exponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the exponent into the equations

Compound Interest
Exponential Growth and Decay
Introduction to Exponential Functions

B.3: demonstrate an understanding of compound interest and annuities, and solve related problems.

B.3.3: determine, through investigation (e.g., using spreadsheets and graphs), that compound interest is an example of exponential growth [e.g., the formulas for compound interest, A = P((1 + i) to the n power), and present value, PV = A((1 + i) to the -n power), are exponential functions, where the number of compounding periods, n, varies]

Compound Interest
Exponential Growth and Decay

C: Trigonometric Functions

C.1: solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications;

C.1.1: solve problems, including those that arise from real-world applications (e.g., surveying, navigation), by determining the measures of the sides and angles of right triangles using the primary trigonometric ratios

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

C.1.2: solve problems involving two right triangles in two dimensions

Sine, Cosine, and Tangent Ratios

C.2: demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;

C.2.3: make connections between the sine ratio and the sine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios, with or without technology (e.g., by generating a table of values using a calculator; by unwrapping the unit circle), defining this relationship as the function f(x) = sinx, and explaining why the relationship is a function

Cosine Function

C.2.4: sketch the graph of f(x) = sinx for angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/ decreasing intervals)

Cosine Function
Sine Function

C.2.5: make connections, through investigation with technology, between changes in a real-world situation that can be modelled using a periodic function and transformations of the corresponding graph (e.g., investigate the connection between variables for a swimmer swimming lengths of a pool and transformations of the graph of distance from the starting point versus time)

Translating and Scaling Sine and Cosine Functions

C.3: identify and represent sine functions, and solve problems involving sine functions, including problems arising from real-world applications.

C.3.3: pose problems based on applications involving a sine function, and solve these and other such problems by using a given graph or a graph generated with technology from a table of values or from its equation

Translating and Scaling Sine and Cosine Functions