Chapter 3: Transformations of Graphs and Data Lesson 2: The Graph-Translation Theorem Mrs. Parziale

Vocabulary: Translation: in a plane, a translation is a transformation that maps each point (x, y) onto (x + h, y + k). Simply put, it is a slide of the graph right or left h units and/or up or down k units

Example 1: Suppose we have a translation given by the rule (x, y)  (x + 1, y + 5) Where does the point P (0, 0) translate to? P’ = _______ Where does the point Q (1, 2) translate to? Q’ = _______ Where does the point R (-8, 4) translate to? R’ = _______ Graph each translation on the coordinate plane to the right.

Example 2: Suppose the point (3, -2) translates to (0, 5) Describe what happened to the x value. Describe what happened to the y value. Under this translation, what would happen to the point (0, 0)? Write the rule for this translation (as written above). (x, y)  Write the rule for a translation of 6 units to the right and 3 units down. (x, y) 

Example 3: Write a rule for translating the point (x,y) 8 units down and 5 right: (x, y)    What kind of translation was this?

Example 4: Given the following two functions: (a) Graph each on the same grid (b) Find the translation rule that maps f  g. (i.e., figure out how f has moved to become g.) (x,y) 

GRAPH TRANSLATION THEOREM The following 2 processes yield the same graph: applying the translation (x, y)  (x + h, y + k) to the graph of the original equation (i.e. translate each point on the graph by this rule) 2. replacing x with (x – h) and y with (y – k) in the original equation **NOTE the minus signs!!!

Example 5: Find an equation for the functions graphed to the right: Steps to follow: Determine the parent function Determine how far the function has moved (from the origin in most cases). Call this h units in the x-direction and k units in the y-direction. Replace (x) with (x-h) and replace y (or f(x)) with (y-k) in the equation. Simplify equation into “y =” form Parent f(x) = Moved Equ’n: Asymptotes?

How about this one? Find an equation for the functions graphed to the right: Steps to follow: Determine the parent function Determine how far the function has moved (from the origin in most cases). Call this h units in the x-direction and k units in the y-direction. Replace (x) with (x-h) and replace y (or f(x)) with (y-k) in the equation. Simplify equation into “y =” form Parent f(x) = Moved Equ’n: Asymptotes?

Try These Parent f(x) = Parent f(x) = Moved Moved Equ’n: Equ’n: Asymptotes? Parent f(x) = Moved Equ’n: Asymptotes?

Try This One Parent f(x) = Moved Equ’n: Asymptotes?

Example 5: Find equations for all asymptotes of the graph of

Example 6: The origin of the graph of y = x3 has translated to the point (2, 3). a. How far have we moved horizontally? b. What is h? c. How far have we moved vertically? d. What is k? e. Write the rule; Substitute (x - ______) into the equation everywhere you see an x, and substitute (y - ______) into the equation everywhere you see a y. g. Simplify this equation into “y =” form. h. Graph this on your graphing calculator. Did (0, 0) translate 2 to the right and 3 up?

Example 8: What if y = x3 is translated so that the origin is now at (-1, 4)? a. What is h? b. What is k? c. Write the rule. d. Write the equation.

Closure What is a translation? Given the graph of y = x2, what would be the equation of the line if the origin translated to point (-2, 4). First, express as y – k = (x – h)2 form. Then, solve for y.