Examples of rotation in the following topics:
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- The four main types of transformations are translations, reflections, rotations, and scaling.
- A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation.
- Where $x_1$and $y_1$are the new expressions for the rotated function, $x_0$ and $y_0$ are the original expressions from the function being transformed, and $\theta$ is the angle at which the function is to be rotated.
- If we rotate this function by 90 degrees, the new function reads:
- Differentiate between three common types of transformations: reflections, rotations, and scaling
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- For objects extended in space, such as a diver jumping from a diving board, the object follows a complex motion as it rotates, while its center of mass forms a parabola.
- Paraboloids are also observed in the surface of a liquid confined to a container that is rotated around a central axis.
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- The shape of the radical graph will resemble the shape of the related exponent graph it were rotated 90-degrees clockwise.
- For example, the graph of $y=\sqrt x$ is the graph of $y=x^2$rotated clockwise.
- Note how it looks like a graph of $y=x^2$rotated clockwise (with the part below the x-axis removed).
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- An arc
may be a portion of a full circle, a full circle, or more than a full
circle, represented by more than one full rotation.
- Because the total
circumference of a circle equals $2\pi$ times the radius, a full circular rotation is $2\pi$ radians.
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- From the graph, it can be seen that the hyperbola formed by the equation $xy = 1$ is the same shape as the standard form hyperbola, but rotated by $45^\circ$.
- Another way to prove it algebraically is to construct a rotated $x$-$y$ coordinate frame.
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- In other words, rotating the graph $180$ degrees about the point of origin results in the same, unchanged graph.
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- Each of the four words in the phrase corresponds to one of the four
quadrants, starting with quadrant I and rotating counterclockwise.