https://mathbooks.unl.edu/OAM/index.php?action=history&feed=atom&title=5.5%3A_Horizontal_Stretches_%26_Compressions 2026-04-04T15:33:02Z Revision history for this page on the wiki MediaWiki 1.32.2 https://mathbooks.unl.edu/OAM/index.php?title=5.5:_Horizontal_Stretches_%26_Compressions&diff=130&oldid=prev 2020-06-01T14:45:20Z

<p>Created page with &quot;<a href="/OAM/index.php/5.4:_Vertical_Stretches_%26_Compressions" title="5.4: Vertical Stretches &amp; Compressions"> Prior Lesson</a> | <a href="/OAM/index.php/5.6:_Combining_Transformations" title="5.6: Combining Transformations"> Next Lesson</a> ==Objectives:== * Recognize that horizontal stretches and compres...&quot;</p> <p><b>New page</b></p><div>[[5.4: Vertical Stretches &amp; Compressions | Prior Lesson]] | [[5.6: Combining Transformations | Next Lesson]]<br /> ==Objectives:==<br /> <br /> * Recognize that horizontal stretches and compressions correspond with changes to the inputs<br /> * Horizontally stretch and compress a function that is given either explicitly or graphically<br /> <br /> ==Important Items==<br /> ===Definitions:=== <br /> horizontal stretch, horizontal compression<br /> <br /> <br /> ----<br /> <br /> ==Lesson Guide==<br /> <br /> Notes to the instructor: The main focus on this section should be taking a given function and knowing how the stretches/compressions affect the graph of this function. <br /> <br /> ===Warm-Up===<br /> <br /> Perhaps remind students again of the transformations they have seen up to this point. Reiterate the ideas of changing inputs and outputs when we apply transformations. <br /> <br /> ===Recognize that horizontal stretches and compressions correspond with changes to the inputs===<br /> <br /> Have students do Problem 1. Discuss the differences of the graphs.<br /> <br /> &lt;!--On the board, draw the graph of a function, $f(x)$, and label some points.<br /> <br /> '''Approach 1''': The Experiment<br /> Pose the following experiment to your students: &quot;What happens if we scale all the inputs by 2?&quot; <br /> (You may want to write this down so that students don't lose track of what you're doing.) <br /> For each point you've labeled, generate a new point that has twice the output value, and draw the new function. Call this function $g(x)$. <br /> Observe with your students that what you've done is a horizontal stretch, and that it'd be nice to be able to find an implicit equation for this. <br /> Point out that you've already described the relationship between $g(x)$ and $f(x)$: The height of $g$ at $x$ is the height of $f$ at $x/2$. You should write this down. <br /> <br /> '''Aproach 2''': The Observation<br /> Tell your students that you'd like to find the graph of $g(x)=f(1/2x)$. <br /> For each point you've drawn, use this equations to find points of g(x). <br /> For example, if (2,4) is a point on f(x), you'd write $g(4)=f((1/2)4)=f(2)=4. <br /> Add this point to the graph. After doing this for all points, ask your students what you've done to the graph physically.<br /> Ask them to observe what has changed, the inputs, or outputs? How have the inputs or outputs changed? <br /> Again, lead them to the following statement that you should write down: The height of $g$ at $x$ is the height of $f$ at $x/2$. You should write this down. <br /> <br /> <br /> Then do your approach again for either dividing the inputs by 2 or by asking them to graph $f(2x)$. Developing something like the following throughout your examples may be helpful:<br /> <br /> <br /> {{#widget:Iframe<br /> |url=https://www.desmos.com/calculator/kgydalgxz4?embed<br /> |width=500<br /> |height=500<br /> |border=0<br /> }}<br /> --&gt;<br /> <br /> <br /> <br /> Observe that the $x$-intercept values change, but the $y$-intercepts stay the same, which makes sense since only the input is being changed.<br /> <br /> ===Horizontally stretch and compress a function that is given either explicitly or graphically===<br /> <br /> Work with students to fill this part out in their course packet (part (d)).<br /> If $f(x)$ is a function and $k&gt;1$ is a constant, then the graph of <br /> * $g(x)=f\left(\frac{1}{k}x\right)$ '''horizontally stretches''' the graph of $f(x)$ by a factor of $k$,<br /> * $g(x)=f(kx)$ '''horizontally compresses''' the graph of $f(x)$ by a factor of $k$.<br /> <br /> If $k&lt;-1$, then the graph of $g(x)$ also involves a reflection of the graph of $f(x)$ about the $y$-axis.<br /> <br /> <br /> Have students do Problems 2-6.<br /> <br /> Have students talk about Problem 7 at their tables. Force each table to make a decision and then appoint someone to write their answer on the board. Use this to lead a discussion.<br /> <br /> ==Comments==<br /> ----</div>

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