Exam 1 - Quadric Surfaces, Common Equations of Motion, and Functions

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Last updated 3:08 PM on 9/18/26
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18 Terms

1
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Determining traces

  • xy trace → set z=0

  • xz trace → set y=0

  • yz trace → set x=0


2
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Ellipsoid

Equation: x2a2+y2b2+z2c2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

Characteristics:

  • All three two-degree terms present

  • All three two-degree terms are positive when the equation equals one

  • All traces are ellipses


<p>Equation: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ </p><p>Characteristics:</p><ul><li><p>All three two-degree terms present</p></li><li><p>All three two-degree terms are positive when the equation equals one</p></li><li><p>All traces are ellipses</p></li></ul><p></p>
3
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Hyperboloid of one sheet

Equation: x2a2+y2b2−z2c2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1

Characteristics:

  • All three two-degree terms are present

  • Two two-degree terms are positive and one is negative when the equation equals one

  • One trace is an ellipse, the other two are hyperbolas

  • The axis is parallel to the negative variable


<p>Equation: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 $$</p><p>Characteristics:</p><ul><li><p>All three two-degree terms are present</p></li><li><p>Two two-degree terms are positive and one is negative when the equation equals one</p></li><li><p>One trace is an ellipse, the other two are hyperbolas</p></li><li><p>The axis is parallel to the negative variable</p></li></ul><p></p>
4
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Hyperboloid of two sheets

Equation: −x2a2−y2b2+z2c2=1-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1

Characteristics:

  • All three two-degree terms are present

  • Two two-degree terms are negative and one is positive when the equation equals one

  • One trace is an ellipse parallel to the third plane (xy plane in the image), two traces are hyperbolas

  • Axis is parallel to the positive variable


<p>Equation: $$ -\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 $$</p><p>Characteristics:</p><ul><li><p>All three two-degree terms are present</p></li><li><p>Two two-degree terms are negative and one is positive when the equation equals one</p></li><li><p>One trace is an ellipse parallel to the third plane (xy plane in the image), two traces are hyperbolas</p></li><li><p>Axis is parallel to the positive variable</p></li></ul><p></p>
5
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Elliptical cone

Equation: x2a2+y2b2−z2c2=0\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0

Characteristics:

  • All three two-degree terms are present

  • Two two-degree terms are positive and one is negative when the equation equals zero

  • Two traces are hyperbolas

  • One trace will be a point or ellipse parallel to the third plane

  • Axis is parallel to the negative variable


<p>Equation: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0 $$</p><p>Characteristics:</p><ul><li><p>All three two-degree terms are present</p></li><li><p>Two two-degree terms are positive and one is negative when the equation equals zero</p></li><li><p>Two traces are hyperbolas</p></li><li><p>One trace will be a point or ellipse parallel to the third plane</p></li><li><p>Axis is parallel to the negative variable</p></li></ul><p></p>
6
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Elliptic paraboloid

Equation: x2a2+y2b2−zc2=0\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z}{c^2}=0

Characteristics:

  • Two two-degree terms are present

  • One one-degree term is present

  • Two traces are parabolas

  • One trace is an ellipse

  • Axis is parallel to the one-degree variable


<p>Equation: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z}{c^2}=0 $$</p><p>Characteristics:</p><ul><li><p>Two two-degree terms are present</p></li><li><p>One one-degree term is present</p></li><li><p>Two traces are parabolas</p></li><li><p>One trace is an ellipse</p></li><li><p>Axis is parallel to the one-degree variable</p></li></ul><p></p>
7
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Hyperbolic paraboloid

Equation: x2a2−y2b2−zc2=0\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z}{c^2}=0

Characteristics:

  • Two two-degree terms are present (one positive and one negative)

  • One one-degree term

  • One trace is a hyperbola

  • Two traces are parabolas

  • Axis is parallel to the one-degree variable


<p>Equation: $$ \frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z}{c^2}=0 $$</p><p>Characteristics:</p><ul><li><p>Two two-degree terms are present (one positive and one negative)</p></li><li><p>One one-degree term</p></li><li><p>One trace is a hyperbola</p></li><li><p>Two traces are parabolas</p></li><li><p>Axis is parallel to the one-degree variable</p></li></ul><p></p>
8
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Sphere

Equation: x2+y2+z2=r2x_{}^2+y^2+z^2=r^2

Characteristics:

  • Three two-degree terms present

  • All traces are circles


<p>Equation: $$ x_{}^2+y^2+z^2=r^2 $$</p><p>Characteristics:</p><ul><li><p>Three two-degree terms present</p></li><li><p>All traces are circles</p></li></ul><p></p>
9
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Cylinder

Equation:x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

Characteristics:

  • If a=ba=b , x2+y2=r2x^2+y^2=r^2 and the trace is a circle

  • General trace is an ellipse

  • Two two-degree terms present, both positive when the equation equals one

  • Axis is parallel to the missing variable


<p>Equation:$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$</p><p>Characteristics:</p><ul><li><p>If $$a=b$$ , $$x^2+y^2=r^2$$ and the trace is a circle </p></li><li><p>General trace is an ellipse</p></li><li><p>Two two-degree terms present, both positive when the equation equals one</p></li><li><p>Axis is parallel to the missing variable</p></li></ul><p></p>
10
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Helix

Equation: r(t)=<rcos⁡(2πNΔtt),rsin⁡(2πNΔtt),ct>r\left(t\right)=<r\cos\left(\frac{2\pi N}{\Delta t}t\right),r\sin\left(\frac{2\pi N}{\Delta t}t\right),ct>

  • The 2πNΔt\frac{2\pi N}{\Delta t} term indicates the rate of revolutions

  • Not a quadric surface


11
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f(x)=mx+bf\left(x\right)=mx+b


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12
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f(x)=x2f\left(x\right)=x^2


13
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f(x)=x3f\left(x\right)=x^3

Graph of y = x cubed, passing through the origin from bottom-left to top-right


14
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f(x)=xf\left(x\right)=\sqrt{x}

Graph of y = square root of x, starting at (0,0) and curving into the first quadrant


15
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f(x)=1xf\left(x\right)=\frac{1}{x}


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16
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f(x)=ln⁡(x)f\left(x\right)=\ln\left(x\right)

Graph of y = ln(x), passing through (1, 0) and increasing slowly


17
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f(x)=exf\left(x\right)=e^{x}


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18
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f(x)=tan⁡(x)f\left(x\right)=\tan\left(x\right)

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