Precalculus Applet Collection
Hands-On demonstrations of Precalculus concepts.
Table of Contents
- Functions, Power Functions, Polynomials
- Review of Functions & Graphs
- Domain of a Function
- Graph Characteristics
- Even & Odd Functions
- Piecewise Functions
- Piecewise Function Application Example
- Graphs of Function Inverses & One-to-One Functions
- Transformations in Baseball
- Function Transformations
- Horizontal Transformations
- Vertical Transformations
- Composition of Functions
- Limits & Continuity
- Understanding "x Approaches c" Notation
- Continuity
- Box - Folding for Maximum Volume
- Polynomial Equations & Inequalities
- Rational Functions
- Rational Function End Behavior
- Exponential and Logarithmic Functions
- The Number "e"
- Trig Ratios and Functions
- Graphs of Trig Functions
- Right Triangles on the Unit Circle
- Graphs of the Sine and Cosine Functions
- Trig Transformations
- Periodicity of Trig Functions
- Circular Diamonds
- The Tangent Function
- Linear and Angular Velocity in Circular Motion
- Restricting the Range of arcsine and arccosine
- Sin(x) and Cos(x) from the Unit Circle (3D)
- Trig Equations and Identities
- Cofunction Identities
- Law of Sines and Cosines with Vectors
- Law of Sines - Ambiguous Case
- Law of Cosines Application
- Law of Sines: No, Right, One, or Two Triangles
- Conic Sections
- Conic Sections
- Conic Sections - General Form
- Conics in Standard Form for Graphing
- Parametric Equations
- Parametch-A-Sketch
- Two Parametric Equations
- Polar Equations
- Polar Coordinates
- Mapping Cartesian Coordiantes to Polar Coordinates
Review of Functions & Graphs
Rational Function End Behavior
The end behavior of a rational function (what [math]y[/math] does as [math]x[/math] grows very large in magnitude) can be determined by the structure of the function's expression. This app demonstrate the three basic cases of horizontal or oblique (slant) asymptote based on the relative degrees of the numerator and denominator polynomials, and their leading coefficients.
The degrees and the leading coefficients of the numerator and denominator polynomials can be set with the four horizontal sliders. Each time a change is made to one of these, a new random function is generated. New functions can also be randomly generated using current slider settings by clicking the [b]New f(x)[/b] button.[br][br][math]f(x)[/math] is graphed in solid red while the HA or OA is graphed in dashed green. The equation of the asymptote is also displayed. Note that the [math]y[/math]-value of the HA or the slope of the OA is determined solely by the leading coefficients, along with the relative degrees, of the numerator and denominator.[br][br]To see how the HA or OA approaches a single line as x grows in magnitude, adjust the Zoom slide to change the scale of the graph. The [math]x[/math] and [math]y[/math] scales are changed simultaneously to maintain the proper aspect ratio of the display.
The Number "e"
The number [b]e[/b], often called the "natural base", is an important constant, just like [math]\pi[/math]. Its value is 2.718 to three decimal places. It is called the natural base because as the base of an exponential function, it models natural growth very well. Here's an app that should give you a little insight into why "[b]e[/b]" is what it is.
Imagine we have a colony of organisms, maybe people or bacteria. At time t=0, we will assume the population is "1", which could mean 1 or 100 or 1 million. After a time interval of 1 (minute, year, millennium), we want this population to double. Our first model has two points: (0,1) and (1, 2). All the action happens in one big burst at time t=1.[br][br]This is not very realistic, since real organisms in large populations tend to reproduce essentially continuously. So let's start moving toward something more continuous. Instead of doubling all at once at t=1, let's increase by half, twice. We'll do the first increase at time t=1/2, when we multiply the population by 1.5. Then, at t=1, we multiply by 1.5 again for the other half. But wait - this means our population increased not by a factor of 2 but by 1.5 x 1.5 = 2.25. Interesting.[br][br]Let's try increasing three times, by 1/3 each time. Now the population increases by 4/3 x 4/3 x 4/3 or about 2.37 times. We can extend this idea to larger numbers of multiplications, at shorter time intervals. Our model becomes[br][br][math]Population=Initial\times(1+\frac{1}{n})^n[/math][br][br]As n approaches infinity, the [math](1+\frac{1}{n})^n[/math] factor asymptotically approaches a value - the number "[i][b]e[/b][/i]"![br][br]On the app, we start with the two points (0,1) and (1, 2). By moving the n slider to larger values, more points appear, representing the increasing number of diminishing increases. The number in the box labeled "m" is the maximum value of n, so you chan change that to larger and larger numbers as you please.[br][br]Notice that the points move toward a curve that ultimately (as n grows toward infinity) becomes the graph of [math]y=e^x[/math]. You can verify this by checking the "Show f(x) = e^x" box. The horizontal red dashed line is at [math]y=e[/math], and so the rightmost point will eventually end up at (1, e), as [math]y=e^1[/math] predicts.
Right Triangles on the Unit Circle
Click and drag the point on the circle to change the right triangle that is formed. Check and uncheck the "Neg [math]\theta[/math]" box to see the effects of measuring [math]\theta[/math] in different directions (counterclockwise and clockwise). See below for more information.
The "[b][u][i]Unit Circle[/i][/u][/b]" is a circle of radius 1, centered at the Origin. We establish the positive x-axis as the "initial side" of an angle, with any ray from the Origin forming its "terminal side".[br][br]If we draw a [b][color=#38761d]segment[/color][/b] from the Origin to the point where the terminal side intersects the Circle, we can define that as the [b][color=#38761d]Hypotenuse[/color][/b] of a right triangle. The triangle's [b][color=#ff0000]Opposite[/color][/b] side will extend from the intersection point vertically (up or down) to the x-axis. Its [b][color=#0000ff]Adjacent[/color][/b] side extends from the Origin to the intersection of the [b][color=#ff0000]Opposite[/color][/b] side and the x-axis.[br][br][b][u]Reference Angles:[/u][/b][br]We define the angle [math]\theta[/math] to extend counterclockwise from the [b]Initial Side[/b] to the [b][color=#38761d]Terminal Side[/color][/b]. However, if we are working with the right triangles described above, the angle in the triangle that we would normally call "[math]\theta[/math]" must be positive and cannot exceed 90 degrees. We define [b]Reference Angles [/b]to be [b]the acute angle from the x-axis to the [/b][color=#38761d][b]HYP[/b][/color]. The [b]Reference Angle[/b] is the "theta" of the right triangle.[br][br][b][u]Coterminal Angles:[/u][/b][br]Since we can continue around the circle as many times as we like, the angle [math]\theta[/math] can exceed 360 degrees ([math]2\pi[/math] radians). However, it is easiest to work with angles in the range of 0 - 360 (or 0 - [math]2\pi[/math] radians). Therefore we define [b]Coterminal Angles [/b]as angles that are at the same place on the circle, but could result from multiple rotations. For example, an angle of 10 degrees is the same as an angle of 370 degrees, since 370 degrees includes one full rotation of 360 degrees plus another 10 degrees. This angle could also be called 730 degrees (2 rotations + 10 deg), 3610 degrees (10 rotations + 10 deg), etc. If [math]n[/math] is the number of rotations, then [math]360n+\theta[/math] (or [math]2\pi n+\theta[/math]) is [i]coterminal with[/i] [math]\theta[/math].[br][br]If we rotate [math]\theta[/math] in the clockwise direction, we consider [math]\theta[/math] to be negative. So, for example, an angle of 330 degrees (clockwise, 330 deg positive) is [i]coterminal with[/i] -30 degrees (counterclockwise, 30 deg negative), since they end at the same point.[br][br][center][b]Co[/b][b]terminal = "sharing the terminal side"[/b][/center][br][b][u]Trig Functions on the Unit Circle:[/u][/b][br]The [b][color=#38761d]Hypotenuse[/color][/b] has a length of 1, since it is the radius of the Unit Circle. Since [math]sin\left(\theta\right)=\frac{OPP}{HYP}[/math], [math]cos\left(\theta\right)=\frac{ADJ}{HYP}[/math], and [b][color=#38761d]HYP[/color]=1[/b], we have the relationships:[br][br][math]sin\left(\theta\right)[/math] = y-coordinate of point at end of [b][color=#38761d]HYP[/color][/b][br][math]cos\left(\theta\right)[/math] = x-coordinate of point at end of [b][color=#38761d]HYP[/color][/b]
Cofunction Identities
The CO function identities relate sine to COsine, tangent to COtangent, and secant to COsecant by COmplemetary angles.
Cofunction Identities
Click the "PLAY" button to see that the two acute angles are always complements.
Law of Sines - Ambiguous Case
The Law of Sines is a formula that can be used to solve all SAA and ASA triangles. It can also be used for SSA triangles, but the triangle resulting from defining angle A and sides a and b depends on the length of side a. If a is too short (a < h), it does not reach the third side c, and no triangle is formed. If a is longer than b, a single triangle results. And if a is between h and b, side a can reach side c in two ways, resulting in two possible triangles.
Law of Sines - Ambiguous Case
The lengths of a and b can be changed in the diagram. Length b can be changed by moving point X, and length a can be changed by adjusting the slider. When a < h, no triangle is formed. When a is between h and b, two possible triangles (shown in red and green) can be formed. When a > b, only one triangle is possible (shown all red).
Conic Sections
[b]Check Out Conic Sections![/b][br][br][list][br][*]Check or Clear "Show Axes" to set visibility of x, y, and z axes[br][/*][*]Adjust Cutting Plane using Tilt (angle), Shift (left/right), and Height (up/down) sliders[br][/*][*]"Reset View" to restore default viewpoint and stop motion[br][/*][*]"Reset Plane" to restore cutting plane to starting orientation[br][/*][*]"Tilt Plane to Slant of Cone" sets plane tangent to cone edge (then adjust Shift or Height for true parabola)[br][/*][*]"View Above Section" sets view directly above (perpendicular to) section[br][/*][*]"Spin It!" to automatically revolve view around the z-axis[br][/*][*]"Stop" to stop revolving[br][/*][*]"Projection Type" button (4th) in 3D view to select 3D glasses[/*][/list][br][br]You can also manipulate the 3D view by [b]Right-Click-Dragging[/b]
Parametch-A-Sketch
Move the sliders to turn the knobs manually. Change the equation in the box next to each knob to have the knobs follow them when you press START. (You should set the sliders to zero when using the automatic drawing feature). Screen scale is restricted to 0-6 for y and 0-9 for x.
Parametch-A-Sketch
Polar Coordinates
Illustration of how points are specified using Polar Coordinates.