Core Focus on Functions & Data Lesson 1.4 Core Focus on Functions & Data Function Translations

Warm-Up If f (x) = 3x2 + 2 and p(x) = − 5, find each value. 1. f (4) 2. p(4) 3. f (−9) 4. p(−2) 50 −4 245

Function Translations Lesson 1.4 Function Translations Perform translations on graphs and equations.

Vocabulary Transformation The movement of a figure on a graph so that it is a different size or in a different position. Parent Function The most basic form of a particular type of function. Translation A transformation in which a figure is shifted up, down, left or right.

Explore! Shifting Graphs Step 1 Set the window of your graphing calculator so that it has a domain of −10 ≤ x ≤ 10 and a range of −10 ≤ y ≤ 10. Graph the parent function y₁ = x² on your calculator. Sketch a graph of this function on your paper. This graph is called a parabola. Step 2 Leaving the parent function in y₁ on your calculator, graph each of the following functions in y₂. Sketch the parent function and the transformation on three separate coordinate planes. a. y₁ = x² b. y₁ = x² c. y₁ = x² y₂ = x² + 4 y₂ = x² − 3 y₂ = x² + 2 Step 3 What happens to the graph of the parent function when you add or subtract a value from the function? Step 4 Without graphing, predict what will happen to the graph of the parent function y = x² when it undergoes the following transformations: a. y = x² − 10 b. y = 6 + x²

Explore! Shifting Graphs Step 5 Leave the parent function in y₁ on your calculator. Graph each of the following functions in y₂. Sketch the parent function and the transformation on three separate coordinate planes. a. y₁ = x² b. y₁ = x² c. y₁ = x² y₂ = (x + 4)² y₂ = (x − 6)² y₂ = (x − 3)² Step 6 What happens to the graph of the parent function when you add or subtract a value from the x-value of the function? Step 7 Predict how the graph of each function will shift compared to the parent function graph. Use your graphing calculator to check your answers. a. y = (x + 5)² b. y = (x − 4)² − 3 c. y = (x + 1)² + 6 Step 8 Write the equation for each a. b. function based on what you learned in this Explore!

Function Translations Translating Up or Down When a is added to the function, the graph shifts up a units. When a is subtracted from the function, the graph shifts down a units. Translating Right or Left When a is subtracted from the x-value of the function, the graph shifts right a units. When a is added to the x-value of the function, the graph shifts left a units.

Example 1 Suppose f (x) is a function. Describe how f (x) is translated in each new function. a. f (x) + 7 b. f (x + 2) − 5 c. 4 + f (x − 1) Seven is being added to the entire function, so the graph will be translated up 7 units. The graph will shift to the left 2 units and down 5 units. The graph is shifted up 4 units and to the right 1 unit.

Example 2 The graph below represents a parent function, g(x). Write a transformation expression for each graph based on its shift(s). a. b. The graph shifts left 1 unit and up 2 units. g(x) The graph shifts down 4 units. A shift of 4 down is shown by subtracting 4 from the function. g(x)  g(x) − 4 A shift to the left of 1 requires 1 to be added to the x-value in the function. A shift up 2 units is shown by adding 2 to the entire function. g(x)  g(x + 1) + 2

Communication Prompt If a geometric figure was translated to a different location on a coordinate plane, is the figure in the end location congruent to the original figure? Why or why not?

Exit Problems Describe the direction and amount of the translation(s) that occurs to the function f(x). 1. f (x) + 3 2. f (x + 2) 3. f (x − 5) − 1 Write an equation for each transformation. 4. Translate the graph right 5 units. 5. Translate the graph f (x) = x2 down two units. up 3 units left 2 units right 5 units, down 1 unit y = | x − 5 | f(x) = x2 − 2