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### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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A parabola is formed when a plane intersects a cone parallel to the side of the cone. This intersection produces only one curve (as compared to two in a hyperbola). On this page, we discuss parabolas in rectangular coordinates. For a discussion of parabolas in polar form, see this separate page.

 Parabola

Depending on the orientation of the cone with respect to the coordinate axis (xy-axis in 2 dimensions), the equations will be different. We will look only at the two cases where the coordinate axes runs parallel to the axis of the cone and perpendicular to the axis of the cone. These two cases will produce four possible parabolas.
1. Parabola opens up.
2. Parabola opens down.
3. Parabola opens to the left.
4. Parabola opens to the right.

We can combine types 1 and 2 to get one equation form and similarly, types 3 and 4 can be combined for a second equation form.

$$(x-h)^2 = 4p(y-k)$$ opens up or down opens left or right

Of course, as with many mathematical equations, there are several ways to write this. However, written this way, we can directly pull a lot of information from it for the graph. (It is easy to convert from the form $$y=ax^2+bx+c$$ for example that you are probably more familiar with, using some algebra and completing the square.)

We use special terms to describe parts of the parabola. You are probably familiar with the vertex. The other attributes are listed below (with equations from a parabola that opens up or down; you should be able to translate them to the other orientation) and shown in Figure 2.

A special comment is in order. Shown very well in this figure are two pink lines with a black bar crossing them. The black bar indicates that the length of these lines are equal to one another. That is the definition that makes this a parabola. In words this says, for every point on the parabola, the distance between the point and the focus is the same as the distance from the point and the perpendicular distance to the directrix.

Classify - - When looking at the equation in the form $$Ax^2+Bxy+Cy^2+$$ $$Dx+Ey+F=0$$, a parabola will have $$B=0$$ and either $$A=0$$ or $$C=0$$ but not both.

$$(x-h)^2 = 4p(y-k)$$

vertex

$$(h,k)$$

axis of symmetry

$$x=h$$

focus

$$(h,k+p)$$

directrix

$$y=k-p$$

$$Ax^2+Dx+Ey+F=0$$

Example $$(x-2)^2=4(y-3)$$

Figure 3 [Built with GeoGebra]

$$(y-k)^2 = 4p(x-h)$$

vertex

$$(h,k)$$

axis of symmetry

$$y=k$$

focus

$$(h+p,k)$$

directrix

$$x=h-p$$

$$Cy^2+Dx+Ey+F=0$$

Example $$(x-2)=-4(y-3)^2$$

Figure 4 [Built with GeoGebra]

### Practice

Sketch the graph of the parabola $$\displaystyle{\frac{x^2}{12}=\frac{y}{3}}$$.

Problem Statement

Sketch the graph of the parabola $$\displaystyle{\frac{x^2}{12}=\frac{y}{3}}$$.

Solution

### 1583 solution video

video by PatrickJMT

Find the focus and the directrix of the parabola $$\displaystyle{y=\frac{2}{5}x^2}$$.

Problem Statement

Find the focus and the directrix of the parabola $$\displaystyle{y=\frac{2}{5}x^2}$$.

Solution

### 1584 solution video

video by PatrickJMT

Write the standard form of the equation for the parabola with the vertex at the origin and the focus at $$(1/8,0)$$.

Problem Statement

Write the standard form of the equation for the parabola with the vertex at the origin and the focus at $$(1/8,0)$$.

Solution

### 1585 solution video

video by PatrickJMT

Write the standard form of the equation for the parabola with the vertex at the origin and the directrix at $$y=-5/6$$.

Problem Statement

Write the standard form of the equation for the parabola with the vertex at the origin and the directrix at $$y=-5/6$$.

Solution

### 1586 solution video

video by PatrickJMT