Skill Descriptor: Analyze transformations of functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Classify transformations applied to graphs or equations of functions
• Describe the effects of transformations by comparing the graph 𝑦 = a𝑓(b(𝑥 − ℎ)) + 𝑘 to the graph of 𝑦 = 𝑓(𝑥)
• Describe transformations of functions using functional notation
• Describe transformations of functions using mathematical language
• Sketch the graph of y = a𝑓(b(𝑥 − ℎ)) + 𝑘 from a given sketch of the function 𝑦 = 𝑓(𝑥) where the equation of 𝑦 = 𝑓(𝑥) is not given
• Write the equation of a function whose graph is transformed from the graph 𝑦 = 𝑓(𝑥)
• Write the equation of a transformed function given descriptions of the transformations applied to 𝑦 = 𝑓(𝑥)

Skill Descriptor: Analyze inverse functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Describe the effects on coordinate points and/or functions when reflected through the line 𝑦 = 𝑥
• Sketch the reflection of 𝑦 = 𝑓(𝑥) through the line 𝑦 = 𝑥 given a graph or equation of 𝑓(𝑥)
• Write the equation of a function given the graph of its reflection through the line 𝑦 = 𝑥
• Explain how the graph of the line 𝑦 = 𝑥 can be used to sketch the inverse of a relation
• Explain how the transformation (𝑥, 𝑦) → (𝑦, 𝑥) can be used to sketch the inverse of a relation
• Justify if the reflection of a function through y = x is a function
• Sketch the graph of the inverse function given the graph of a function
• Identify the domain restrictions of a function so its inverse is a function
• Determine the inverse function given a graph y = 𝑓(𝑥)
• Determine the inverse function algebraically given the equation y = 𝑓(𝑥)
• Explain relationships between 𝑓(𝑥) and the inverse function of 𝑓(𝑥)
• Identify if two functions are inverses of each other algebraically and/or graphically including functions previously covered and functions within Pre-Calculus 120A
• Apply properties of inverses to the graphs of trigonometric functions

Skill Descriptor: Analyze radical functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Identify the characteristics of radical functions including domain, range, and intercepts
• Sketch the graph of the function $y=a\sqrt{b(x-h)}+k$ by applying transformations to the graph of the function $y=\sqrt{x}$
• Explain why the domain and range of the function 𝑦 = 𝑓(𝑥), and its square root function $y=\sqrt{x}$ , may differ
• Determine radical expressions from given graphs
• Generalize characteristics between radical functions and their indices

Skill Descriptor: Analyze exponential functions.

Global Competencies: CTPS

Achievement Indicators:

• Identify characteristics of exponential functions including domain, range, intercepts, and equations of horizontal asymptotes
• Sketch the graph of an exponential function by applying transformations to the graph
• Determine exponential equations that represent given graphs

Skill Descriptor: Analyze logarithmic functions.

Global Competencies: CTPS

Achievement Indicators:

• Identify characteristics of logarithmic functions including domain, range, intercepts, and equations of asymptotes
• Sketch the graph of the function  $y=a{\log }_c\left(b(x-h)\right)+k$
  by applying transformations to the graph of the function $y={\log }_{c}x$
• Determine logarithmic equations that represent given graphs

Skill Descriptor: Evaluate logarithmic and exponential expressions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Change expressions from logarithmic form to exponential form and vice versa
• Determine the exact value of a logarithm without technology
• Justify the estimated value of a logarithm using benchmarks
• Approximate the values of natural logarithms or powers base 𝑒 using technology

Skill Descriptor: Apply logarithmic laws to solve problems.

Global Competencies: CTPS

Achievement Indicators:

• Generalize the laws for logarithms using numeric examples and exponent laws
• Expand logarithmic expressions with variable and numerical inputs using logarithmic laws
• Condense logarithmic expressions with variable and numerical inputs using logarithmic laws
• Write equivalent logarithmic expressions with different bases

Skill Descriptor: Solve problems that involve exponential and logarithmic equations.

Global Competencies: CM, CTPS

Achievement Indicators:

• Determine solutions of exponential equations in which the expressions can be rewritten with the same integer base
• Determine solutions of exponential equations in which the bases are not powers of one another
• Explain why a value obtained in solving logarithmic equations may be extraneous
• Model phenomena with exponential functions
• Interpret characteristics of functions within given post-secondary, work ready, or life contextual problems
• Solve exponential growth and decay problems
• Solve financial application problems involving exponential equations
• Solve problems involving 𝑒^x and ln x

Skill Descriptor: Represent angles in standard position.

Global Competencies: CTPS

Achievement Indicators:

• Sketch angles in standard position (positive or negative)
• Sketch angles in standard position with measures expressed in the form 𝑘 π radians where 𝑘 is a rational number
• Convert angle measurements between radians (exact value or decimal approximation) and degrees
• Determine the reference angle of given angle measurements
• Determine coterminal angles within given domains
• Determine general form of all coterminal angles given angles
• Solve problems using the relationship between the radian measure of an angle in standard position and the length of the arc cut on a circle of radius r

Skill Descriptor: Develop the equation of the unit circle.

Global Competencies: CTPS

Achievement Indicators:

• Derive the equation of the unit circle from the Pythagorean Theorem
• Describe the six trigonometric ratios using a point 𝑃(𝑥, 𝑦) that is the intersection of the terminal arm of an angle and the unit circle
• Generalize the equation of a circle with centre (0, 0) and radius 𝑟, and circles with centres not at the origin and radius 𝑟

Skill Descriptor: Solve problems using trigonometric ratios for angles.

Global Competencies: CM, CTPS

Achievement Indicators:

• Determine the approximate value of a trigonometric ratio for any angle with technology
• Determine the exact value of a trigonometric ratio for special angles
• Justify strategies used to determine the exact value of trigonometric ratios
• Determine angles in a specified domain given the value of a trigonometric ratio
• Determine the exact values of trigonometric ratios given the coordinates of a point on the terminal arm of an angle in standard position
• Determine the angles in a specified domain given a point on the terminal arm of an angle in standard position
• Determine the exact values of trigonometric ratios given the value of one trigonometric ratio in a specified domain

Skill Descriptor: Solve problems using sine, cosine, tangent functions.

Global Competencies: CM, CTPS

Achievement Indicators:

• Sketch the graph of 𝑦 = sin 𝑥, 𝑦 = cos 𝑥, and 𝑦 = tan 𝑥
• Determine the characteristics of 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑, 𝑦 =𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑, or 𝑦 = tan 𝑥 given a graph
• Determine the characteristics of 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑, 𝑦 =𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑, or 𝑦 = tan 𝑥 given an equation
• Describe how transformations affect the graphs of sinusoidal functions
• Sketch graphs of the form 𝑦 = 𝑎 sin 𝑏(𝑥 − 𝑐) + 𝑑 or 𝑦 =𝑎 cos 𝑏(𝑥 − 𝑐) + 𝑑 without technology
• Determine the sine and cosine functions that represent given graphs
• Model cyclical phenomena using sinusoidal functions
• Analyze graphs of trigonometric functions to solve problems

Skill Descriptor: Solve trigonometric equations.

Global Competencies: CTPS

Achievement Indicators:

• Verify if a given value is a solution to a trigonometric equation
• Solve first-degree trigonometric equations (i.e., one or more steps to solve)
• Solve second-degree trigonometric equations
• Determine the approximate solution(s) of first- and second-degree trigonometric equations with a restricted domain using technology
• Determine the approximate general solution of given first- and second-degree trigonometric equations using technology

Skill Descriptor: Prove trigonometric identities algebraically.

Global Competencies: CM, CTPS

Achievement Indicators:

• Verify trigonometric identities numerically
• Explain why verifying the two sides of a trigonometric identity are equal for given values is insufficient to conclude the identity is valid
• Determine the non-permissible values of a trigonometric identity using known trigonometric identities
• Prove algebraically that a given trigonometric identity is valid
• Determine the exact value of a trigonometric ratio using sum, difference, or double-angle identities for given angles