Precalculus Term 2
This is a collection of sketches used with my 2nd term Precalculus class, taught to juniors at Stuyvesant High School.
Table of Contents
- Conic Sections
- Drawing a Parabola
- Locus Definition of a Parabola
- Parabola Properties
- Parabola: Focal Chord
- Parabola: Tangent Line
- Parabola: Tangents to the Endpoints of a Focal Chord
- Parabola: Reflective Property
- Parabola: Reflective Property and Distance
- Ellipse: Definition
- Ellipse: Graphing
- Ellipses and Parabolas: Eccentricity
- Ellipse: Reflective Property
- Ellipse: Reflective Property and Distance
- Whispering Gallery - Grand Central
- Hyperbola - Graphing
- Hyperbola: Proof of Reflective Property
- Hyperbola: Reflective Property Demonstration
- Conic Sections
- EC: Hyperbolas for Location
- Hyperbolas: Reflective Property and Distance
- Complex Numbers
- Nth Roots of a Complex Number
- Multiplying Complex Numbers in Polar Form
- DeMoivre's Theorem: Visualization
- Polar Form of Complex Numbers
- Polar Coordinates
- Polar Coordinates
- Polar: Distance Formula
- Circle Centered at (r₀,θ₀) with Radius d
- Tangent Line to a Circle
- Graph Polar Functions
- Graphing Limaçons
- Graphing Lemniscates
- Rose Curves
- Lemniscate Envelope
- Parametric Equations
- Graphing Parametric Equations
- Combining Parametric Equations
- Cycloid: Derivation
- Cycloid: Adjusting the Radius
- Cycloid Demonstration
- Epicycloid
- Deriving the Epicycloid Equations
- Hypocycloid
- Deriving the Hypocycloid Equations
- Projectile Motion
- Parametric Equations - Point on a Basketball
- Matrices
- Intersections of Planes
- Determinant of a 2x2 Matrix
- Linear Transformations with Matrices
- Limits
- Oscillating Function
- Slope of the Tangent Line
- Limit of Sin(x) at Infinity
Drawing a Parabola
Place Points on the graph that are equidistant from F and the given line.
Nth Roots of a Complex Number
Use the sketch above to plot the n[sup]th[/sup] roots of a complex number. You can zoom in and out using the zoom slider.[br]Type in whatever modulus and argument that you choose as well as the index for the root.
Polar Coordinates
Use the sliders to adjust the radius and angle for [math]P[/math].[br]Change the Grid to polar and add the Circle and Line to see how we are graphing. Basically, we want to first find the angle, then use the radius to either go in the direction of the angle (if [math]r>0[/math]) or in the opposite direction (if [math]r<0[/math])
Graphing Parametric Equations
Graph parametric equations by entering them in terms of [math]t[/math] above. You can set the minimum and maximum values for [math]t[/math]. Pay attention to the initial point, terminal point and direction of the parametric curve.[br]
Intersections of Planes
You can use this sketch to graph the intersection of three planes. Simply type in the equation for each plane above and the sketch should show their intersection.[br][br]The lines of intersection between two planes are shown in orange while the point of intersection of all three planes is black (if it exists)[br][br]The original planes represent a dependent system, with the orange line as the solution.
Oscillating Function
Shown is the graph of [math]f(x)=\sin\frac{1}{x}[/math][br][br]This sketch demonstrates why the limit of this function does not exist at 0. The function oscillates between -1 and 1 increasingly rapidly as [math]x\rightarrow0[/math]. In a way you can think of the period of oscillation becoming shorter and shorter. Click the "Zoom In" button to see what happens as we get closer to [math]x=0[/math]. The graph becomes so dense it seems to fill the entire space. For this reason, the limit does not exist as there is no single value that the function approaches.[br][br][center][math]\lim_{x\to0}f\left(x\right)[/math] does not exist.[/center]