This lesson gives practice with all the graphical transformations:
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.
| EQUATION | ACTION | GRAPHICAL RESULT (WORDS) |
GRAPHICAL RESULT (cumulative graph) |
COMMENTS | |||||
| $y = x^2$ | the ‘basic model’ | (the starting graph) |
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start with the graph of the squaring function | |||||
| $y = (x-1)^2$ | replace every $\,x\,$ by $\,x-1\,$ | right $\,1\,$ |
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when applying a transformation that changes the $\,x\,$-values of points, use the phrase ‘replace every $\,x\,$ by ...’ |
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| $y = 3(x-1)^2$ | multiply previous $\,y\,$-values by $\,3\,$ |
vertical stretch, $\,(a,b) \mapsto (a,3b)\,$ |
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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 $\,3\,$ |
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| $y = 3(x-1)^2 + 5$ | add $\,5\,$ to previous $\,y\,$-values | up $\,5\,$ |
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Other orders will work.
For example:
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PRACTICE: Given the Desired Graphical Results, Produce the Equations
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Give the equations and actions that produce the successive desired graphical results. Check your answers by clicking the ‘SHOW ANSWER’ buttons. |
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| EQUATION | ACTION | GRAPHICAL RESULT | |
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
even and odd functions
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
even and odd functions