L4 Transformations - Part 1
Completion requirements
Unit 1A
Precalculus
Lesson 4: Transformations
What do droplets of water from a water fountain have in common with the trajectory of a basketball bouncing off the floor?
A bouncing ball captured with a stroboscopic flash at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»23«/mn»«/mstyle»«/math» images per second.
All objects on Earth are subjected to the force of gravity. When an object goes up in the air, eventually the force of gravity will act on it and pull it towards the ground. Galileo was the first scientist to show that the height of an object launched
into the air follows a path in the shape of a parabola.
Consider a ball thrown upward into the air with a velocity of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mrow»«/mstyle»«/math». The height of the ball is modelled by the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»d«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»10«/mn»«mi»t«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»d«/mi»«/mstyle»«/math» is the height above the ground in metres and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» is the time in seconds. This height is frequently referred to as displacement, which is the position of an object relative to where it began.
Consider a ball thrown upward into the air with a velocity of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mrow»«/mstyle»«/math». The height of the ball is modelled by the function «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»d«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»10«/mn»«mi»t«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»d«/mi»«/mstyle»«/math» is the height above the ground in metres and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»t«/mi»«/mstyle»«/math» is the time in seconds. This height is frequently referred to as displacement, which is the position of an object relative to where it began.
As you can see from the graph, the displacement of the ball resembles the basic parabola «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math»,
with a few changes. To transition from the basic parabola to one modelling the ball’s vertical displacement, it is necessary to perform a number of transformations (translations, reflections, and stretches) on the basic parabolic function «math
style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math».
There are five different types of transformations that are commonly applied to functions.
- Vertical Stretches
- Horizontal Stretches
- Reflections
- Vertical Translations
- Horizontal Translations
