Honors Algebra 2
This book contains sketches created for my Honors Algebra 2 class at Stuyvesant High School
Table of Contents
- Trigonometry
- Unit Circle Trigonometry
- Unit Circle: Complementary Angles
- Unit Circle: Sine and Cosine Functions
- Graph of Cosine
- Graphing Sine/Cosine with Translations
- Graphing the Tangent Function
- Lighthouse Function p.553 #69
- Cosine of the Difference of Two Angles
- The Ambiguous Case of SSA
- Complex Numbers
- Multiplying Complex Numbers
- Exponential and Logarithmic Functions
- Exponential graph as 2-variable function
- Conic Sections
- Drawing a Parabola
- Locus Definition of a Parabola
- Parabola Properties
- Parabola: Reflective Property and Distance
- Systems of Equations
- Intersection of a Parabola and a Line
- Intersection of Two Circles
- Intersection of a Circle and Parabola
- Intersection of Two Parabolas
- Intersections of Planes
- Statistics and Probability
- Normal Distribution - Data Collection
- Normal Distribution
- Sample Means
- Distribution of Sample Means
- Sample Means 1
- Sample Means 2
- Tossing a fair coin n times
- Comparing Normal and Binomial Distributions
Unit Circle Trigonometry
Drag the slider for [math]\theta[/math] to adjust the angle. Notice the coordinates of point [math]P[/math] on the unit circle.
Multiplying Complex Numbers
This graph shows how we can interpret the multiplication of complex numbers geometrically.[br]Given two complex numbers:[br][math]w=a+bi[/math], [math]z=c+di[/math] where [math]a,b,c,d\in\mathbb{R}[/math][br][br]Consider their product[br][math]wz=\left(a+bi\right)z=az+biz[/math][br][table][tr][td]1[/td][td][math]az[/math][/td][td]Dilate [math]z[/math] by a scale factor of [math]a[/math][/td][/tr][tr][td]2[/td][td][math]iz[/math][/td][td]Rotate [math]z[/math] by [math]90^\circ[/math] about [math]0[/math][/td][/tr][tr][td]3[/td][td][math]biz[/math][/td][td]Dilate [math]iz[/math] by a scale factor of [math]b[/math][/td][/tr][tr][td]4[/td][td][math]az+biz[/math][/td][td]Translate [math]az[/math] by [math]biz[/math][/td][/tr][/table][br]Focus on the two right triangles in the diagram:[br][list=1][*]The right triangle formed by [math]w[/math], [math]0[/math] and the positive real axis.[/*][*]The right triangle formed by [math]az[/math], [math]biz[/math] and [math]0[/math][br][br][/*][/list]The first right triangle has sides of length: [math]a[/math], [math]b[/math], [math]\left|w\right|[/math].[br]The second right triangle has sides of length [math]a\left|z\right|[/math], [math]b\left|z\right|[/math], and [math]\left|wz\right|[/math].[br][br]Since we have the proportion: [math]\frac{a\left|z\right|}{a}=\frac{b\left|z\right|}{b}[/math], we can conclude the triangles are similar since two pairs of corresponding sides are proportional and their included angles (the right angles) are congruent.[br][br]This has two implications:[br][list=1][*]The ratio of similitude is [math]\left|z\right|[/math], which means that [math]\left|wz\right|=\left|w\right|\left|z\right|[/math] (this is an alternative to the algebraic proof you did for homework)[/*][*]The angle formed by [math]wz[/math], [math]0[/math] and [math]az[/math] is congruent to [math]Arg\left(w\right)[/math], since they are corresponding angles of similar triangles[/*][/list][br]This leads us to our other conclusion, that [math]Arg\left(wz\right)=Arg\left(z\right)+Arg\left(w\right)[/math][br][br]Key results:[br][math]\left|wz\right|=\left|w\right|\left|z\right|[/math][br][math]Arg\left(wz\right)=Arg\left(w\right)+Arg\left(z\right)[/math]
Exponential graph as 2-variable function
Drawing a Parabola
Place Points on the graph that are equidistant from F and the given line.
Intersection of a Parabola and a Line
To see how many ways we can intersect a line and a parabola, try adjusting the red line above to see how many times it can meet the blue circle.[br][br]Drag points [math]A[/math] and [math]B[/math] around to change the location and slope of the red line.
What is the maximum number of points of intersection?