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TACKLING TRANSLATIONS Translating Graphs of Functions - shifting a graph along the x or y-axis (vertically or horizontally). Which variable controls changes along the x-axis? Which variable controls changes along the y-axis? How do we translate graphs? - The input of a function is the variable that is used to find the output of an equation. The input is usually found in the equation. - The output of a function is the result of plugging a number into an equation. The output is usually found by itself on one side of the equation and has been solved for in terms of the input variable. ***The output of a function is represented by f(input variable) which is said to be “f of a” if a were the input variable. Ex: An equation of a line in slope-intercept form is:_____________ What is the input of this equation? What is the output of this equation? Write the equation of a line in the form of a function:_____________ *Therefore, f(x) represents the variable ___ when graphing a function. In order to shift a graph along either axis, we need to replace the variables with different “instructions”. A. Applying translating to equations: If we add to the x-variable (or input), the graph will move ________. If we subtract from the x-variable (or input), the graph will move ________. If we add to the y-variable (or output), the graph will move ________. If we subtract from the y-variable (or output), the graph will move ________. B. Applying translating to a function: To translate a function in the form f(x), we need to replace the input and the output with different “instructions”. Direction Number Variable f(x+2) will move the graph _______ for ______ on the _______ axis. f(x-9) will move the graph _______ for ______ on the _______ axis. f(x)-8 will move the graph _______ for ______ on the _______ axis. f(x)+3 will move the graph _______ for ______ on the _______ axis. Write the directions for translating a graph up 3 and left 2: __________________ C. Writing the equation for a translated function: 1) When translating a function or equation of a line, we are first given an equation or function to translate. f(x) = x + 5 2) y=x+5 Next, we are given instructions to tell us how the graph will move. Left 2, Up 3 3) or or f(x + 2) - 3 Then, we replace the input and output (or variables) with the proper instructions. (y – 3) = (x + 2) + 5 4) Finally, we solve for the output variable. y–3=x+2+5 y–3=x+7 y–3+3=x+7+3 y = x + 10 Bonus Page: What is unique about a line with a slope of one when it is translated? (Hint: Use your calculator to graph an equation. Then, translate it. Which axis did it move on?) What happens if you want to translate a line with slope 1 along both the x and y-axis the same distance? (Hint: Graph y = x and translate it the same distance on both the x and y axis.) Why doesn’t this work if the slope is not 1? Why doesn’t this work if the graph is not a line?