Pre-Calculus 12 Pacing Guide - This pacing guide replaces the previous yearly plan. It has been updated to reflect removed outcomes and provide flexibility for responsive instruction.
Pre-calculus 12 Desmos Activity Collection - A collection of online student Desmos activities organized by unit.
EAL Support - Desmos offers a free suite of math software tools, including the Desmos Graphing Calculator and Scientific Calculator, as well as free digital classroom activities. Click on the globe in the tool bar to access the site in other languages.
RF02 Students will be expected to demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of functions and their related equations. [C, CN, R, V]
RF02.01 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.
RF02.02 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.
RF02.03 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.
RF02.04 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.
RF02.05 Write the equation of a function whose graph is a vertical and/or horizontal translation of the graph of the function y = f(x).
RF03 Students will be expected to demonstrate an understanding of the effects of horizontal and vertical stretches on the graphs of functions and their related equations. [C, CN, R, V]
RF03.01 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.
RF03.02 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.
RF03.03 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.
RF03.04 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.
RF03.05 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).
RF04 Students will be expected to apply translations and stretches to the graphs and equations of functions. [C, CN, R, V]
RF04.01 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.
RF04.02 Write the equation of a function, given its graph that is a translation and/or stretch of the graph of the function y = f(x).
RF05 Students will be expected to 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, and line y = x. [C, CN, R, V]
RF05.01 Generalize the relationship between the coordinates of an ordered pair and the coordinates of the corresponding ordered pair that results from a reflection in the x-axis, the y-axis, or the line y = x.
RF05.02 Sketch the reflection of the graph of a function y = f(x) in 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.
RF05.03 Generalize, using inductive reasoning, and explain rules for the reflection of the graph of the function y = f(x) in the x-axis, the y-axis, or the line y = x.
RF05.04 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.
RF05.05 Write the equation of a function, given its graph that is a reflection of the graph of the function y = f(x) in the x-axis, the y-axis, or the line y = x.
RF06 Students will be expected to demonstrate an understanding of inverses of relations. [C, CN, R, V]
RF06.01 Explain how the graph of the line y = x can be used to sketch the inverse of a relation.
RF06.02 Explain how the transformation (x, y) ⇒ (y, x) can be used to sketch the inverse of a relation.
RF06.03 Sketch the graph of the inverse relation, given the graph of a relation.
RF06.04 Determine if a relation and its inverse are functions.
RF06.05 Determine restrictions on the domain of a function in order for its inverse to be a function.
RF06.06 Determine the equation and sketch the graph of the inverse relation, given the equation of a linear or quadratic relation.
RF06.07 Explain the relationship between the domains and ranges of a relation and its inverse.
RF06.08 Determine, algebraically or graphically, if two functions are inverses of each other.
Additional Resources and Activities for RF02, RF03, RF04 and RF05 (function transformations):
Filling a Swimming Pool - A handout for students to think about function transformation in terms of a swimming pool slowly filling over time.
Function Transformation Card Matching Activity - A set of cards with equations, graphs and descriptions. Students worked alone or with a partner (their choice) to match the appropriate description, equation, and graph cards together into 16 sets of three. Read more about this activity from Chris Rime.
Graphing Piecewise Functions in Desmos - An example Desmos graph showing a piecewise function with transformations applied.
Transformation Word Search Activity - *Updated June 2025* A self-checking activity where the graph of f(x) is provided and when transformations of f(x) are graphed, the transformed graph connects four letters on that coordinate plane. When these four letters are unscrambled, they form a common (school-appropriate) four-letter word.
Additional Resources and Activities for RF06 (inverse functions):
Exploring Inverse Functions investigation - Ask student to evaluate a function (such as f(x) = (x-3)/2) and plot points to create a graph. Then swap the x and y values and plot the points again to create a graph of g(x). Fold the graph paper so that f(x) and g(x) coincide. Ask students how the two graphs are related geometrically. Try again with other functions to see if this relationship holds.
Inverse Functions Lesson - A google slides introducing functions and inverses as input/output machines. Several practice questions at the end.
Inverses, Domains and Ranges - Students find functions, inverses, domains and ranges in order to fill in the empty spaces on a table in this Word document. Original from kirbatron on TES.
Inverse Functions and Logarithms - An exploration of function inverses. Created by Julie Reulbach.
Find the Inverse of a Function Foldable - *Updated April 2025* (From Sarah Carter's blog M+A+T+H=LOVE)