![The graph of $ f( x)$ is given below. Use it to graph the
following.
\begin{alignat*}{4}
&a)\; f( {3x}) &\qquad\qquad &b)\; f( {-x}) &\qquad\qquad &c)\; f( {-2x}) &\qquad\qquad
&d)\; f( {x-1})\\
&e)\; f( {x}) + 1 &\qquad\qquad &f)\;5 f( {x}) &\qquad\qquad &g)\; f( {x+2}) &\qquad\qquad
&h)\;5 f( {3x+2}) + 1
\end{alignat*}
i) In your own words, describe the manner in which the graph of $ f( x)$
changes when we: multiply $ f( x)$ by a constant; add a constant to $ f( x)$;
multiply $x$ by a constant; add a constant to $x$.
j) Describe the process of drawing $ u(x) = a f( {bx+c}) + d,$ where
$a,\;b,\;c,\;$ and $d$ are constants when given only the graph of $ f( x)$.
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\centerline{\includegraphics[width=5truein]{ws3p1jf93.pdf}}](/latex/cache-iuLqVG58Wp8zDxqTeHOzZA/srender-crop.png) | • pdf • pdf tight • latex and fig 1 |
Tags: transformations, graphing-intro, composite, piecewise-defined, abstract-function, illustrative-example, sketching, g--a--g, a--g--v, qualitative, t2, c1
I like this problem a lot. By using a non-trivial function it allows students to see what happens to a function under various transformations. In addition, it allows students to generalize the concepts that they have formed from the concrete examples that they just created. The only downside to this problem, is that it takes a while to finish. Be prepared to spend time on this one.
– Darrin Visarraga 1998-08-09
Surprisingly, students often do not have a good sense for how transformations/compositions change a parent function. This problem will take students at least thirty minutes to an hour to complete. They get a feel for how the transformations affect the shape of the original “parent” function and then they are asked to describe the effect. With technology today, one could incorporate a problem preceding this one that has them investigate transformations of a particular function, say f(x)=
| x | + x^3 and then have them move into this problem. |