Match each equation with its polar graph from choices A–D.
r = cos 2θ

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Recall that the equation given is in polar form: $r = \cos 2\theta$. This is a type of rose curve, which generally has the form $r = \cos n\theta$ or $r = \sin n\theta$.

Identify the parameter $n$ in the equation. Here, $n = 2$, which means the rose curve will have \$2n$ petals if $n$ is even, so it will have 4 petals.

Understand the shape: For $r = \cos 2\theta$, the petals are symmetrically placed around the origin, with petals aligned along the angles where $\cos 2\theta$ reaches its maximum values.

To match the graph, look for a polar plot with 4 petals evenly spaced around the origin, each petal corresponding to the angles where $\cos 2\theta = 1$ or $-1$.

Compare the given graph options A–D to this description, selecting the one that shows a 4-petal rose curve centered at the origin with petals aligned along the axes.

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Polar Coordinates and Graphs

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how equations like r = cos 2θ plot points based on θ is essential to visualize and match the graph correctly.

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05:32

Intro to Polar Coordinates

Rose Curves and Their Properties

Equations of the form r = cos(nθ) or r = sin(nθ) produce rose curves with petals. When n is even, the curve has 2n petals; when n is odd, it has n petals. Recognizing this helps identify the shape and number of petals in the graph.

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3:37

Roses

Symmetry in Polar Graphs

Polar graphs like r = cos 2θ exhibit symmetry about the polar axis or other lines. Understanding symmetry properties aids in matching the equation to its graph by predicting the orientation and repetition of petals.

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Percorso guidato