12.6 Figures/Equation Cards

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Note: The negative variable dictates the direction the figure opens with the exception of a Hyperbolic Paraboloid

Last updated 9:27 PM on 9/20/26
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9 Terms

1
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Hyperboloid of One Sheet

Equation: (x²/a²) - (y²/b²) + (z²/c²) = 1

  • Note: For this figure, y is negative. This results in the figure rotated sideways as seen in the image


<p>Equation: (x²/a²) - (y²/b²) + (z²/c²) = 1</p><ul><li><p>Note: For this figure, y is negative. This results in the figure rotated sideways as seen in the image</p></li></ul><p></p>
2
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Hyperboloid of One Sheet

Equation: (x²/a²) +(y²/b²) - (z²/c²) = 1

  • Note: In this figure, z is negative. This results in the hyperboloid opening in the z direction


<p>Equation: (x²/a²) +(y²/b²) - (z²/c²) = 1</p><ul><li><p>Note: In this figure, z is negative. This results in the hyperboloid opening in the z direction</p></li></ul><p></p>
3
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Elliptic Cone

Equation: (x²/a²) + (y²/b²) - (z²/c²) = 0

  • Note: When y = 0, the cross section of the figure is an X


<p>Equation: (x²/a²) + (y²/b²) - (z²/c²) = 0</p><ul><li><p>Note: When y = 0, the cross section of the figure is an X</p></li></ul><p></p>
4
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Hyperboloid of Two Sheets

Equation: (x²/a²) + (y²/b²) - (z²/c²) = -1

  • Note: The difference between two sheets and one sheet is in what the equation is equal to. When equal to -1, the resulting figure is of two sheets.


<p>Equation: (x²/a²) + (y²/b²) - (z²/c²) = -1</p><ul><li><p>Note: The difference between two sheets and one sheet is in what the equation is equal to. When equal to -1, the resulting figure is of two sheets.</p></li></ul><p></p>
5
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Elliptic Paraboloid

Equation: (x²/a²) + (y²/b²) - (z/c) = 0

  • Note: For c > 0, the figure opens in the positive. For c < 0, the figure opens in the negative


<p class="is-empty is-editor-empty">Equation: (x²/a²) + (y²/b²) - (z/c) = 0</p><ul><li><p class="is-empty is-editor-empty">Note: For c &gt; 0, the figure opens in the positive. For c &lt; 0, the figure opens in the negative</p></li></ul><p></p>
6
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Hyperbolic Paraboloid

Equation: (x²/a²) - (y²/b²) - (z/c) = 0

  • Note: In other instances, you would multiple the equation by -1 to get one negative and two positives, but since only two of the three variables are quadratic and set equal to 0, you need not multiply by -1.


<p>Equation: (x²/a²) - (y²/b²) - (z/c) = 0</p><ul><li><p>Note: In other instances, you would multiple the equation by -1 to get one negative and two positives, but since only two of the three variables are quadratic and set equal to 0, you need not multiply by -1.</p></li></ul><p></p>
7
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Plain old Cylinder

Equation: x² + y² = 1

  • Note: Cylinders only have two variables, as they are 2D curves with lines going through them all around to create a 3D shape


<p>Equation: x² + y² = 1</p><ul><li><p>Note: Cylinders only have two variables, as they are 2D curves with lines going through them all around to create a 3D shape</p></li></ul><p></p>
8
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Parabola

Equation: y = x²

<p>Equation: y = x²</p>
9
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Ellipse

Equation: x² + 4z² = 4

  • Note: The two variables included in the equation dictate the plane in which the ellipse lies


<p>Equation: x² + 4z² = 4</p><ul><li><p>Note: The two variables included in the equation dictate the plane in which the ellipse lies</p></li></ul><p></p>