MCR 3U1 Investigation: Name: Lessons 2.3 - 2.5: Transformations of Functions For this investigation, use: https://www.desmos.com/calculator/do51gfawwr For each part of the investigation, the appropriate equations for each of the required functions have been pre-populated in Desmos. You will just need to find them and select them. *Remember to "un-select" them for each section of the investigation. Translations Transformations that result in a shift {horizontal (right/left) or vertical (up/down)} on the original function without changing its shape. Complete the table for the remaining functions. The first one is modelled for you. Function Parent Function |Horizontal Translation Vertical Translation General Function Notation f (x ) g(x) = f(x-d) g (x) = f(x)+ c Quadratic f (x) = x2 g (x) = (x - d)2 g ( x ) = x2 + c Radical f (x) = Vx Rational f ( x) = = Absolute Value f ( x ) = 1x1 Example: Apply the following transformations on each of the parent functions using Desmos. Write the equation of the new function, g(x), in each case. g (x) = f(x - 2) g(x) = f(x+ 2) 9 (x ) = f (x ) - 2 9 ( x) = f (x ) + 2 g ( x ) = g (x ) = 9 ( x ) = 9 ( x ) = g ( x ) = g (x ) = g (x ) = 9 ( x ) = g (x ) = g (x ) = g (x ) = g ( x ) = g ( x ) = g (x) = 9 ( x ) = g (x ) = What happens to the key points? In particular, how do the values of d and c affect the x and y coordinates?Stretches / Compressions Transformations that result in vertical or horizontal stretches or compressions of the original function. Complete the table for the remaining functions. The first one is modelled for you. Function Parent Function Vertical Horizontal Stretch/Compression Stretch/Compression General Function Notation f (x) af (x) f ( kx ) Quadratic f (x ) = x2 g (x ) = ax2 g (x ) = (kx) 2 Radical f (x ) = Vx Rational f (x ) = - Absolute Value f ( x ) = 1x1 Example: Apply the following transformations on each of the parent functions using Desmos. Write the equation of the new function, g(x), in each case. g (x) = 2f (x) g (x ) = =f(x) g (x) = f(2x) g (x) = f (x) g (x ) = g (x ) = g (x ) = g ( x ) = g (x ) = g (x) = 9 ( x ) = g (x ) = g (x ) = g (x ) = g (x ) = g (x ) = g ( x ) = g ( x ) = g (x ) = g (x ) = NOTE: Vertical Stretch when: Horizontal Stretch when: Vertical Compression when: Horizontal Compression when: What happens to the key points? In particular, how do stretches/compressions affect the x and y coordinates? Summary: The following will give the effects of all the transformations on the x and y coordinates. Mapping Notation: Function Notation:Reflections Transformations that result in a reflection of the original function over the x or y axis. Complete the table for the remaining functions. The first one is modelled for you. Function Parent Function |Reflection over x-axis Reflection over y-axis General Function Notation f(x ) g (x) = -f(x) g(x) =f(-x) Quadratic f (x ) = x2 g (x ) = -x2 g (x ) = (-x) 2 Radical f (x ) = Vx Rational f ( x( ) = Absolute Value f ( x ) = 1x1 Example: Apply the following transformations on each of the parent functions using Desmos. g (x ) = -f(x) 9 (x ) = f(-x) What happens to the key points? In particular, how do reflections affect the x and y coordinates