Download Caroline Smith MAE 301 - Class Notes for Wednesday 12/8/10 The

Caroline Smith
MAE 301 - Class Notes for Wednesday 12/8/10
The final exam will potentially include the following topics:
 Probability /combinatory (1 or 2 questions)
 Isometry / translations / reflections / rotations (as they relate to high school math)
 Equivalence relations
 Extending a definition to another field
 Complex numbers (relating math to teaching)
 Rational functions / rational numbers
 Distinguish between number sets as related to groups / fields / rings
 Trig functions / inverse trig functions
 Injective / surjective functions (in relation to high school math)
From the homework:
For Y=1/X, rotate the graph through -/4 and write the equation of the resulting graph.
Y=1/X
X² - Y² = 2
Note that the new graph is not a function of x.
Its equation is f(x) = y =  (x²-2)
This can also be written as the general form of this hyperbola x² - y² = 2
Conics:
Conic sections include
 Hyperbola
 Parabola
 Circle
 Ellipse
Why are these called conic sections?
Because they are slicing a cone in two sections.
We defined a degenerate conic as the intersection of two lines or a point.
Algebraically, to be a conic, an equation must be a two degree polynomial. There are two
variables in this polynomial, (x,y) of degree two. With two variables in degree two there
are potentially six terms. ax²+by²+cxy+dx+ey+f
Define:
Ellispe - the set of points such that the sum of the distances from any point to each of the
two fixed points (foci) is a constant.
Circle - set of points equidistant from the center fixed point.
Hyperbola - the difference of the distances from any point to each of the fixed points
(foci) is constant.
High School Standard equations:
Circle: (x-a)² + (y-b)² = r² or x² + y² = r²
Hyperbola: x² - y² = 2 or (x/2)² - (y/2)² = 1
Ellipse: (x/a)² + (y/b)² = 1
Questions:
1. Given the equation (x/a)² + (y/b)² = 1, can you find the foci?
H=(b² + x²)
(x + a) + (a - x)
2(b² + x²) = 2a
b² + x² = a²
x² = a² - b²
2. Given a diagram of a conic section, can you derive the equation?
Use arbitrary point (x,y) to see if you can get to (x/a)² + (y/b)² = 1 using the pythagorean
theorem.