Lesson Hyperbola (concept and graphing calc.)

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This Lesson (Hyperbola (concept and graphing calc.)) was created by by Nate(3500) About Me : View Source, Show
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Hyperbolas, as you most likely already know, are two identical parabolas opening in opposite directions. There are two main ones known as: horizontal and vertical. The transverse axis is a straight line that connects the two parabolas. The distance of the transverse axis is defined as 2a. The conjugate axis is the axis perpendicular to the transverse axis and goes through the center. This axis's distance is defined as 2b. This makes an imaginery box which can be used to better help those who draw hyperbolas on a grid by freelance.
Standard Format: 1+=+%28x+-+h%29%5E2%2Fa%5E2+-+%28y+-+k%29%5E2%2Fb%5E2 ~> For a horizontal transverse axis.
Standard Format: 1+=+%28y+-+k%29%5E2%2Fa%5E2+-+%28x+-+h%29%5E2%2Fb%5E2 ~> For a vertical transverse axis.
Center: (h,k)
First, we will work with the standard form for a horizontal transverse axis.
1+=+%28x+-+h%29%5E2%2Fa%5E2+-+%28y+-+k%29%5E2%2Fb%5E2
1+-+%28x+-+h%29%5E2%2Fa%5E2+=+-%28y+-+k%29%5E2%2Fb%5E2
-1+%2B+%28x+-+h%29%5E2%2Fa%5E2+=+%28y+-+k%29%5E2%2Fb%5E2
b%5E2%28-1+%2B+%28x+-+h%29%5E2%2Fa%5E2%29+=+%28y+-+k%29%5E2
+-sqrt%28b%5E2%28-1+%2B+%28x+-+h%29%5E2%2Fa%5E2%29%29+=+sqrt%28%28y+-+k%29%5E2%29
+-b%2Asqrt%28%28-1+%2B+%28x+-+h%29%5E2%2Fa%5E2%29%29+=+y+-+k
+-b%2Asqrt%28%28-1+%2B+%28x+-+h%29%5E2%2Fa%5E2%29%29+%2B+k+=+y
+-b%2Asqrt%28%28-a%5E2%2Fa%5E2+%2B+%28x+-+h%29%5E2%2Fa%5E2%29%29+%2B+k+=+y
+-b%2Asqrt%28%28-a%5E2+%2B+%28x+-+h%29%5E2%29%2Fa%5E2%29+%2B+k+=+y
+-%28b%2Fa%29%2Asqrt%28-a%5E2+%2B+%28x+-+h%29%5E2%29+%2B+k+=+y
Ironically, +-b/a is the slope for the asymptotes for this hyperbola.
Asymptote for horizontal hyperbolas:
y - y1 = m(x - x1) for the center (h,k)
y - k = (+-b/a)(x - h)
which is: y - k = (b/a)(x - h) and y - k = (-b/a)(x - h)
y - k = bx/a - bh/a and y - k = -bx/a + hb/a
y = bx/a - bh/a + k and y = -bx/a + hb/a + k
Now, lets see how this works:
Center: (2,1)
Transverse Axis: 4 horizontally ~> 2a = 4 or a = 2
Conjugate Axis: 6 vertically ~> 2b = 6 or b = 3
+-%28b%2Fa%29%2Asqrt%28-a%5E2+%2B+%28x+-+h%29%5E2%29+%2B+k+=+y
+-%283%2F2%29%2Asqrt%28-4+%2B+%28x+-+2%29%5E2%29+%2B+1+=+y
Asymptote:
y = bx/a - bh/a + k and y = -bx/a + hb/a + k
y = 3x/2 - 3(2)/2 + 1 and y = -3x/2 + (2)3/2 + 1
y = 3x/2 - 3 + 1 and y = -3x/2 + 3 + 1
y = 3x/2 - 2 and y = -3x/2 + 4
Now, the graphing:
graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C-%283%2F2%29%2Asqrt%28-4+%2B+%28x+-+2%29%5E2%29+%2B+1%2C%283%2F2%29%2Asqrt%28-4+%2B+%28x+-+2%29%5E2%29+%2B+1%2C-3x%2F2+%2B+4%2C3x%2F2+-+2%29
It looks successful....
Now, we will work with the standard form for a vertical transverse axis.
1+=+%28y+-+k%29%5E2%2Fa%5E2+-+%28x+-+h%29%5E2%2Fb%5E2
1+%2B+%28x+-+h%29%5E2%2Fb%5E2+=+%28y+-+k%29%5E2%2Fa%5E2
a%5E2%281+%2B+%28x+-+h%29%5E2%2Fb%5E2%29+=+%28y+-+k%29%5E2
sqrt%28a%5E2%281+%2B+%28x+-+h%29%5E2%2Fb%5E2%29%29+=+sqrt%28%28y+-+k%29%5E2%29
+-a%2Asqrt%28%281+%2B+%28x+-+h%29%5E2%2Fb%5E2%29%29+=+y+-+k
+-a%2Asqrt%28%281+%2B+%28x+-+h%29%5E2%2Fb%5E2%29%29+%2B+k+=+y
+-a%2Asqrt%28%28b%5E2%2Fb%5E2+%2B+%28x+-+h%29%5E2%2Fb%5E2%29%29+%2B+k+=+y
+-a%2Asqrt%28%28b%5E2+%2B+%28x+-+h%29%5E2%29%2Fb%5E2%29+%2B+k+=+y
+-%28a%2Fb%29%2Asqrt%28b%5E2+%2B+%28x+-+h%29%5E2%29+%2B+k+=+y
Ironically again, +-a/b is the slope for the asymptotes for this hyperbola.
Asymptote for vertical hyperbolas:
y - y1 = m(x - x1) for center point (h,k)
y - k = (+-a/b)(x - h)
y - k = (a/b)(x - h) or y - k = (-a/b)(x - h)
y - k = ax/b - ah/b or y - k = -ax/b + ah/b
y = ax/b - ah/b + k or y = -ax/b + ah/b + k
Now, lets see how this works:
Center: (-2,-3)
Transverse Axis: 8 vertically ~> 2a = 8 or a = 4
Conjugate Axis: 6 horizontally ~> 2b = 6 or b = 3
+-%28a%2Fb%29%2Asqrt%28b%5E2+%2B+%28x+-+h%29%5E2%29+%2B+k+=+y
+-%284%2F3%29%2Asqrt%289+%2B+%28x+%2B+2%29%5E2%29+-+3+=+y
Asymptote:
y = ax/b - ah/b + k or y = -ax/b + ah/b + k
y = 4x/3 - 4(-2)/3 - 3 or y = -4x/3 + 4(-2)/3 - 3
y = 4x/3 + 8/3 - 9/3 or y = -4x/3 - 8/3 - 9/3
y = 4x/3 - 1/3 or y = -4x/3 - 17/3
Now, the graphing:
graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C%284%2F3%29%2Asqrt%289+%2B+%28x+%2B+2%29%5E2%29+-+3%2C%28-4%2F3%29%2Asqrt%289+%2B+%28x+%2B+2%29%5E2%29+-+3%2C-4x%2F3+-+17%2F3%2C4x%2F3+-+1%2F3%29

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