Transcript Conic Sections

Conic Sections - Lines
• General Equation:
• y – y1 = m(x – x1)
• M = slope, (x1,y1) gives one point on this line.
• Standard Form:
• ax + by = c
• Basic Algebra can be used to change the format of a
line from point-slope (general equation) to standard
form.
• Special Cases:
• Parallel Lines: Will have the same slope.
• Perpendicular Lines: Will have a negative reciprocal
slope. Ex – one slope of 2/3 other slope of -3/2
Conic Sections – Parabolas
• General Equation:
•
•
•
•
y – y1 = a(x-x1)2
ONLY 1 VALUE CAN BE SQUARED IN A PARABOLA!
(x1,y1) now gives turning point of the parabola.
“a” value determines opening direction
– a value positive: parabola opens up
– a value negative: parabola opens down
• “a” value also determines parabola behavior
– a value is 1: parabola is normal
– a value stronger than 1: parabola is skinny
– a value fraction weaker than 1: parabola is wide
Conic Sections – Parabolas – Cont’d
• Other format for parabola:
• y = ax2 + bx + c
• formula x = -b/(2a)
–Gives the x value of the parabola’s turning
point
–To find the y value of turning point, plug x
value into equation
• Use x value of turning point to balance table on
graphing calculator
• Graph points to model parabola
Conic Sections - Circles
• Both x and y must be squared!
• Anytime x and y are both squared, must be algebraically
converted to equal to 1.
• General Equation:
• (x – x1)2 + (y - y1)2
a2
a2
= 1
• A and b must be the same number for a circle
• (x1,y1) gives the center point of the circle
• “a” value gives the radius of the circle
Conic Sections - Ellipses
• Both x and y must be squared!
• Anytime x and y are both squared, must be algebraically
converted to equal to 1.
• General Equation:
• (x – x1)2 + (y - y1)2
a2
b2
= 1
• a and b must be the DIFFERENT numbers for an ellipse
• (x1,y1) gives the center point of the ellipse
• “a” value gives the x- radius / x-stretch of the ellipse
• “b” value gives the y-radius / y-stretch of the ellipse
Conic Sections - Hyperbolas
• Both x and y must be squared! MUST HAVE MINUS SIGN!!
• Anytime x and y are both squared, must be algebraically
converted to equal to 1.
• General Equation: Either x or y can come first.
• (x – x1)2 - (y - y1)2 = 1
a2
b2
• a and b can be the same or different, doesn’t matter for
hyperbola
• (x1,y1) gives the center point – a and b still give “Stretch
values” just like ellipse.
• Hyperbola breaks at stretch points, depending on formula.