MultiStep Practice with all the Graphical Transformations
This lesson gives practice with all the graphical transformations:
 Shifting graphs up/down left/right
 Horizontal and vertical stretching/shrinking
 Reflecting about axes, and the absolute value transformation
Before practicing the multistep exercises on this page, you should practice Singlestep transformations, all mixed up. Jump to the exercises at the bottom, and clickclickclick to check your understanding. When you're successfully answering all the question types, then you're ready for this current lesson!
Example: Graphing Using Transformations
Graph $\,y = 3(x2)^2 + 5\,$ by starting with a ‘basic model’ and successively applying graphical transformations.
(Here's the entire example in a single image! But it's too small to see easily—so keep reading ... )
Equation:  $y = x^2$ 
Action:  the ‘basic model’ 
Graphical Result (words):  (the starting graph) 
Graphical Result (cumulative graph): 

Comments:  Start with the graph of the squaring function 
Equation:  $y = (x1)^2$ 
Action:  replace every $\,x\,$ by $\,x1\,$ 
Graphical Result (words):  right $\,1\,$ 
Graphical Result (cumulative graph): 

Comments:  When applying a transformation that changes the $x$values of points, use the phrase: ‘Replace every $\,x\,$ by ...’ 
Equation:  $y = 3(x1)^2$ 
Action:  multiply previous $y$values by $\,3\,$ 
Graphical Result (words):  vertical stretch, $\,(a,b) \mapsto (a,3b)\,$ 
Graphical Result (cumulative graph): 

Comments: 
When applying a transformation that changes the $y$values of points, use the language: ‘Do something to previous $y$values’ The notation ‘$\,(a,b) \mapsto (a,3b)\,$’ is read as: ‘$\,(a,b)\,$ maps to $\,(a,3b)\,$’ It clearly shows that the $y$values of previous points are getting multiplied by $\,3\,.$ 
Equation:  $y = 3(x1)^2 + 5$  
Action:  add $\,5\,$ to previous $y$values  
Graphical Result (words):  up $\,5\,$  
Graphical Result (cumulative graph): 

Comments: 
Other orders will work. For example:

PRACTICE: Given the Desired Graphical Results, Produce the Equations
Give the Equations and Actions that produce the successive desired Graphical Results. Check your answers by clicking the ‘Show Answer’ (SA) buttons. 













