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Figure 7.1.5(a) can be drawn by applying the following to the equation of the cardioid. Clicking on the graph shown below will bring up the Plot Builder with all these options already selected. However, executing the complete worksheet with the !!! button in the toolbar will delete the graph drawn by the Plot Builder. If that's done, restore the original condition of the worksheet by closing the corrupted one and relaunching it.
Context Panel: Plots≻Plot Builder
2D implicit plot
coordinates: polar
axis coordinates: polar
$r$ $\to$0 to 4
$\mathrm{\θ}$ $\to$0 to 2*Pi
2D Options:
grid refine $\to$ 1



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$r\=2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)$$\to$
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Alternatively, use the
.

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In the first pane of the Interactive Plot Builder, make the selections shown in the upper portion of Figure 7.1.5(b). The Interactive Plot Builder defaults to an implicit plot.

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Change the ranges for $r$ and $\mathrm{\θ}$ to those shown in Figure 7.1.5(b).

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Then, click on the Options button, and in the lowerleft corner change the coordinate system to the option shown in the lower portion of Figure 7.1.5(b).

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The resulting graph will be Figure 7.1.5(a).



Figure 7.1.5(b) Polar graph via the Plot Builder






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Alternatively, execute either of the first two commands in Table 7.1.5(a) to obtain Figure 7.1.5(a). To obtain a graph of the cardioid on a rectangular Cartesian grid, execute either of the second two commands in the table. (Select Evaluate in the Context Panel.)
$\mathrm{plot}\left(2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)\,\mathrm{\θ}\=0..2\mathrm{pi;}comma;\mathrm{coords}equals;\mathrm{polar}comma;\mathrm{axiscoordinates}equals;\mathrm{polar}comma;\mathrm{scaling}equals;\mathrm{constrained}\right)$

$\mathrm{plots}:\mathrm{polarplot}\left(2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)\,\mathrm{\θ}\=0..2\mathrm{pi;}comma;\mathrm{scaling}equals;\mathrm{constrained}\right)$

$\mathrm{plots}:\mathrm{implicitplot}\left(r\=2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)\,r\=0..4\,\mathrm{\θ}\=0..2\mathrm{pi;}comma;\mathrm{coords}equals;\mathrm{polar}comma;\mathrm{gridrefine}equals;3comma;\mathrm{scaling}equals;\mathrm{constrained}\right)$

$\mathrm{plot}\left(2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)\,\mathrm{\θ}\=0..2\mathrm{pi;}comma;\mathrm{coords}equals;\mathrm{polar}comma;\mathrm{scaling}equals;\mathrm{constrained}\right)$

Table 7.1.5(a) Commands that will generate a graph of the cardioid $r\=2\left(1\mathrm{cos}\left(\mathrm{\θ}\right)\right)$



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To graph the Cartesian equivalent of the cardioid, use the following command. (Select Evaluate in the Context Panel.)
$\mathrm{plots}:\mathrm{implicitplot}\left(\sqrt{{x}^{2}\+{y}^{2}}\=2\left(1\frac{x}{\sqrt{{x}^{2}\+{y}^{2}}}\right)\,x\=4..1\,y\=3..3\,\mathrm{gridrefine}\=3\,\mathrm{scaling}equals;\mathrm{constrained}\right)$



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Context Panel: Plots≻2D Implicit Plot≻$x\,y$ applied to the Cartesian equation will also graph the cardioid.
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