Student Actions

Teacher Actions

• Develop a deep and flexible conceptual understanding of how vertical and horizontal translations affect a graph and its table of values for linear and absolute value functions.

• Develop the ability to make conjectures, model, and generalize by exploring graphs and corresponding tables of values for parent functions f(x) = x and f(x) = |x| and extending patterns to determine how horizontal and vertical transformations affect properties of these parent functions.

• Develop the ability to communicate mathematically when discussing, writing, interpreting, and translating ideas of how transformations of both linear and absolute value functions will have the same behaviors in how the functions are translated. 

• Facilitate meaningful mathematical discourse by providing students with opportunities to compare and contrast graphs of the parent functions and their transformed graphs and communicate their thinking in a variety of ways (verbal description, visual model, and/or written explanation).

• Pose purposeful questions that encourage students to notice patterns in the graphs and tables of the parent functions before and after transformation. 

Key Understandings

Misconceptions 

• Translations of the parent function of linear or absolute value can be represented graphically and within the equation.

• The use of graphing utilities can help students visualize the changes that various translations make in the graph of a function.

• Adding a positive or negative value to the end of a function (i.e. f(x)+c) causes the function to vertically shift in the direction and same distance of the c value.

• Adding a positive or negative value to x in a function (i.e. f(x+c) ) causes the function to horizontally shift in the opposite direction and the same distance as the c value. 

• Students may mistake a horizontal translation for a vertical translation or vice versa. 

• Students may assume that a horizontal translation/shift moves the direction and distance of the value added to x like with vertical translations without recognizing that they move in the opposite direction but the same distance of the value added to x. 

• Students may shift a function in the wrong direction for a horizontal shift. For example, a student may think that f(x)=(x+3) is a horizontal translation of 3 units to the right rather than to the left. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Predict the effects on the graph of a linear function with a change in slope or the y-intercept of the function (PA.A.2.4) 

• Predict the effects of transformations on exponential, radical, quadratic and logarithmic functions (A2.F.1.2)