Multi-Step Practice with all the Graphical Transformations

Index cards 12.5 thru 15

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 multi-step exercises on this page, you should practice Single-step transformations, all mixed up. Jump to the exercises at the bottom, and click-click-click 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 $\,y = 3(x-2)^2 + 5\,$ 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:	$y = (x-1)^2$
Action:	replace every $\,x\,$ by $\,x-1\,$
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:	$y = 3(x-1)^2$
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:	$y = 3(x-1)^2 + 5$
Action:	add $\,5\,$ to previous $y$-values
Graphical Result (words):	up $\,5\,$
Graphical Result (cumulative graph):
Comments:	Other orders will work. For example: $y = x^2$ (entire sequence, lightest to darkest) $y = 3x^2$ $y = 3x^2 + 5$ $y = 3(x-1)^2 + 5$

PRACTICE: Given the Desired Graphical Results, Produce the Equations