$\"\"$

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Rotation formula:

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If the $\"\"$ and $\"\"$ -axis are rotated through an angle $\"\"$ , the coordinates $\"\"$ of

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a point $\"\"$ relative to the $\"\"$ - plane and the coordinates $\"\"$ of the same point

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relative to the new $\"\"$ and $\"\"$ -axis and are related by the formulas

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$\"\"$ and $\"\"$ .

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The general form is $\"\"$

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The angle is $\"\"$ .

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$\"\"$

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The equation is $\"\"$ .

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Compare it with $\"\"$

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$\"\"$ and $\"\"$ .

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The angle is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$

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$\"\"$

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Since $\"\"$ . the angle lies in second quadrant.

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$\"\"$

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$\"\"$

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Rotation of $\"\"$ -axis $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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Rotation of $\"\"$ -axis $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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The rotation formulas are $\"\"$ and $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$ $\"\"$

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Complete the square.

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$\"\"$

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This equation is the standard form of the parabola.

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$\"\"$

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Graph:

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(1) Draw the coordinate plane.

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(2) Draw the rotated coordinate plane.

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(3) graph the function $\"\"$ .

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$\"\"$

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$\"\"$

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The angle is $\"\"$

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The parabola equation is $\"\"$ .

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The graph the parabola with rotated axis.

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$\"\"$