## Topic outline

- General
- Welcome to Pre-Calculus 12
### Welcome to Pre-Calculus 12

The following are resources for students and parents to refer to throughout the semester.

- PLO: B1
### PLO: B1

Demonstrate an understanding of operations on, and compositions of, functions.

Write a function h(x) as the sum, difference, product or quotient of two or more functions.

Determine the value of the composition of functions when evaluated at a point (from an equation and a graph, including: f (f(a)), f(g(a)) and g(f(a)) .

Determine, given the equations of two functions f(x) and g(x), the equation of the composite

function: f ( f (x)), f (g ( x)) and g ( f ( x)) and explain any restrictions.Write a function h(x) as the composition of two or more functions.

Write a function h(x) by combining two or more functions through operations on, and

compositions of, functions.**ASSESSMENT: PLO: B1 Friday Sept.15/17****RE-ASSESSMENT:** - PLO: B2
### PLO: B2

Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations.

Compare the graphs of a set of functions of the form y – k = f (x) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effect of k.

Compare the graphs of a set of functions of the form y = f (x − h) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effect of h.

Compare the graphs of a set of functions of the form y − k = f (x − h) to the graph of y = f (x), and generalize, using inductive reasoning, a rule about the effects of h and k.

Sketch the graph of y − k = f (x) , y = f (x − h) or y − k = f (x − h) for given values of h and k, given a 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 a vertical and/or horizontal translation of the graph of the function y = f (x) .

**Practice Questions****ASSESSMENT:** - PLO: B3
### PLO: B3

Demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations.

Compare the graphs of a set of functions of the form y = af (x) to the graph of y = f (x) , and

generalize, using inductive reasoning, a rule about the effect of a.

Compare the graphs of a set of functions of the form y = f (bx) to the graph of y = f (x) , and

generalize, using inductive reasoning, a rule about the effect of b.

Compare the graphs of a set of functions of the form y =af (bx) to the graph of y = f (x) ,

and generalize, using inductive reasoning, a rule about the effects of a and b.

Sketch the graph of y = af (x) , y = f (bx) or y =af (bx) for given values of a and b, given a

sketch of the function y = f(x), where the equation of y = f (x) is not given.

Write the equation of a function, given its graph which is a vertical and/or horizontal stretch

of the graph of the function y = f (x) .**Practice Work****Assessment** - PLO: B4
### PLO: B4

Apply translations and stretches to the graphs and equations of functions.

Sketch the graph of the function y − k = af (b(x − h)) for given values of a, b, h and k, given

the graph of the function y = f (x) , where the equation of y = f (x) is not given.

Write the equation of a function, given its graph which is a translation and/or stretch of the

graph of the function y = f (x) .**Practice Questions****Assessment** - PLO: B5
### PLO: B5

Demonstrate an understanding of the effects of reflections on the graphs of functions and their related equations, including reflections through the:

*x*-axis,*y*-axis, line*y*=*x*.Generalize the relationship between the coordinates of an ordered pair and the coordinates of

the corresponding ordered pair that results from a reflection through the x-axis, the y-axis or

the line y = x.Sketch the reflection of the graph of a function y = f (x) through the x-axis, the y-axis or the

line y = x, given the graph of the function y = f (x) , where the equation of y = f (x) is not

given.Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the

function y = f (x) through the x-axis, the y-axis or the line y = x.Sketch the graphs of the functions y = −f (x), y = f (−x) and x = −f (y), given the graph of the

function y = f (x), where the equation of y = f (x) is not given.Write the equation of a function, given its graph which is a reflection of the graph of the

function y = f (x) through the x-axis, the y-axis or the line y = x.**Practice Questions** - PLO: B6
### PLO: B6

Demonstrate an understanding of inverses of relations.

Explain how the transformation (x, y) => (y, x) can be used to sketch the inverse of a relation.

Sketch the graph of the inverse relation, given the graph of a relation.

Determine if a relation and its inverse are functions.

Determine restrictions on the domain of a function in order for its inverse to be a function.

Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.

Explain the relationship between the domains and ranges of a relation and its inverse.

Determine, algebraically or graphically, if two functions are inverses of each other.

Assessments:

- PLO: B9
### PLO: B9

Graph and analyze exponential and logarithmic functions.

Practice:

- PLO: B7 & B8
### PLO: B7 & B8

Demonstrate an understanding of logarithms including the product, quotient and power laws of logarithms.

Practice:

Assessment:

- PLO: B10
### PLO: B10

Solve problems that involve exponential and logarithmic equations.

**Practice:****Assessments:** - PLO: B13
### PLO: B13

Graph and analyze radical functions (limited to functions involving one radical).

Assessment:

- PLO: B11
### PLO: B11

Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

**Assessment:** - PLO: B12
### PLO: B12

Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5 )

Assessment:

- PLO: A1
### PLO: A1

Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

**Practice:****Assessment:** - PLO: A3
### PLO: A3

Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.

**Practice:****Assessment:** - PLO: A4
### PLO: A4

Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.

**Practice:****Assessments:** - PLO: A5
### PLO: A5

Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.