Integer intersections and tangents
Integer intersections and tangents
Intersection lines with parabolas and tangents
Parabolas with one intersection
Kissing or mutual tangent a parabola with a maximum and one with a minimum
Kissing or mutual tangent parabolas with different size
Cubic curve with a horizontal bend tangent
The orthogonal hyperbola
Integer intersections and tangents
y = 3x + 2 is in ( –1 , –1) the tangent at y = x2 + 5x + 3
y = 3x + 3 intersects y = x2 + 5x + 3 in (–2 , –3) and (0 , 3)
y = 3x + 6 intersects y = x2 + 5x + 3 in (–3 , –3) and (1 , 9)
y = 3x + 11 intersects y = x2 + 5x + 3 in (–4 , –1) and (2 , 17)
y = 3x + 18 intersects y = x2 + 5x + 3 in (–5 , 3) and (3 , 27)
y = –3x – 13 is in ( –4 , –1) the tangent at y = x2 + 5x + 3
y = –3x – 12 intersects y = x2 + 5x + 3 in (-5 , 3) and (-3 , -3)
y = –3x – 9 intersects y = x2 + 5x + 3 in (-6 , 9) and (-2 , -3)
y = –3x – 4 intersects y = x2 + 5x + 3 in (-7 , 17) and (-1 , -1)
y = –3x + 3 intersects y = x2 + 5x + 3 in (-8 , 27) and ( 0 , 3)
y = 3x – 13 is in (4 , -1) the tangent at y = x2 – 5x + 3
y = 3x – 12 intersects y = x2 – 5x + 3 in ( 3 , –3) and ( 5 , 3)
y = 3x – 9 intersects y = x2 – 5x + 3 in ( 2 , –3) and ( 6 , 9)
y = 3x + 3 intersects y = x2 – 5x + 3 in ( 0 , 3) and ( 8 , 27)
y = 3x + 12 intersects y = x2 – 5x + 3 in (–1 , 9) and ( 9 , 39)
y = 3x + 23 intersects y = x2 – 5x + 3 in (–2 , 17) and (10 , 53)
y = –3x – 19 is in (4 , –7 ) the tangent at y = x2 – 5x + 3
y = –3x –18 intersects y = x2 – 5x + 3 in (3 , –9) and (5 , –3)
y = –3x –10 intersects y = x2 – 5x + 3 in (1 , –7) and (7 , 11)
y = –3x – 3 intersects y = x2 – 5x + 3 in (0 , –3) and (8 , 21)
y = –3x – 6 intersects y = x2 – 5x + 3 in (–1 , 3) and (9 , 33)
y = 3x – 4 is in (1 , -1) the tangent at y = x2 + x – 3
y = 3x – 3 intersects the parabola in ( 0 , –3) and (2 , 3)
y = 3x intersects the parabola in (-1 , –3) and (3 , 9)
y = 3x + 5 intersects the parabola in (-2 , –1) and (4 , 17)
y = 3x + 12 intersects the parabola in (-3 , 3) and (5 , 27)
y = –3x – 7 is in (–2 , –1) the tangent at y = x2 + x – 3
y = –3x – 6 intersects the parabola in (–3 , 3) and (–1 , –3)
y = –3x – 3 intersects the parabola in (–4 , 9) and ( 0 , –3)
y = –3x + 2 intersects the parabola in (–5 , 17) and ( 1 , –1)
y = –3x + 9 intersects the parabola in (–6 , 27) and ( 2 , 3)
y = 2x – 6 is in (3 , 0) the tangent at y = x2 – 4x + 3 = (x – 1)(x – 3)
y = 2xx – 5 intersects the parabola in (2 , –1) and ( 4 , 3)
y = 2x – 2 intersects the parabola in ( 1 , 0) and ( 5 , 8)
y = 2x + 3 intersects the parabola in ( 0 , 3) and ( 6 , 15)
y = 2x +10 intersects the parabola in (–1 , 8) and ( 7 , 24)
y = –2x – 19 is in (–2 , –15) the tangent at y = x2 + 2x – 15 = (x – 3)(x + 5)
y = –2x – 18 intersects the parabola in (–3 , –12) and (–1 , –16)
y = –2x – 15 intersects the parabola in (–4 , – 7) and ( 0 , –15)
y = –2x – 10 intersects the parabola in (–5 , 0) and ( 1 , –12)
y = –2x – 3 intersects the parabola in (–6 , 9) and ( 2 , –7)
Go up
br> Choose the parabola y = x2 – 4x + 3 = (x – 1)(x – 3)
In (–3 , 24) the tangent is y = –10x – 6
In (–2 , 15) the tangent is y = – 8xx – 1
In (–1 , 8) the tangent is y = – 6x + 2
In ( 0 , 3) the tangent is y = – 4x + 3
In ( 1 , 0) the tangent is y = – 2x + 2
In ( 2 , –1) the tangent is y = – 1
In ( 3 , 0) the tangent is y = 2x – 6
In ( 4 , 3) the tangent is y = 4x – 13
In ( 5 , 8) the tangent is y = 6x – 22
In ( 6 , 15) the tangent is y = 8x – 33
In ( 7 , 24) the tangent is y = 10x – 46
Choose the parabola y = x2 – 4x – 21 = (x + 3)(xx – 7)
In (–3 , 0) the tangent is y = –10x – 30
In (–2 , –9) the tangent is y = – 8x – 25
In (–1 , –16) the tangent is y = – 6x – 22
In ( 0 , –21) the tangent is y = – 4x – 21
In ( 1 , –24) the tangent is y = – 2x – 22
In ( 2 , –25) the tangent is y = – 25
In ( 3 , –24) the tangent is y = 2x – 30
In ( 4 , –21) the tangent is y = 4x – 37
In ( 5 , –16) the tangent is y = 6x – 46
In ( 6 , – 9) the tangent is y = 8x – 57
In ( 7 , 0) the tangent is y = 10x – 70
Parabolas with one intersection
Looking for the intersection the square term cancels.
Go up
parabola 1 parabola 2 intersect at y = x2 – 16x + 60 y = x2 – 8x + 12x ( 6 , 0 ) y = x2 – 14x + 40 y = x2 + 2x – 8 ( 3 , 7 ) y = x2 – 14x + 56 y = x2 – 7x ( 8 , 8 ) y = x2 – 13x + 20 y = x2 – 10x + 29 ( –3 , 68 ) y = x2 – 13x + 30 y = x2 – x – 42 ( 6 , – 12 ) y = x2 – 13x + 42 y = x2 + x ( 3 , 12 ) y = x2 – 12x + 35 y = x2 – 7x + 10 ( 5 , 0 y = x2 – 12x + 42 y = x2 – 6x ( 7 , 7 ) y = x2 – 11x + 10 y = x2 – 5x + 4 ( 1 , 0 ) y = x2 – 10x + 21 y = x2 – 9x + 18 ( 3 , 0 ) y = x2 – 1x + 22 y = x2 – 9x + 21 ( 1 , 13 ) y = x2 – 9x + 7 y = x2 – 5x – 1 ( 2 , –7 ) y = x2 – 8x + 22 y = x2 – 5x + 19 ( 1 , 15 ) y = x2 – 7x + 6 y = x2 – 5x ( 3 , –6 ) y = x2 – x6x y = x2 – x – 20 ( 4 , –8 ) y = x2 – 6x + 8 y = x2 + x – 6 ( 2 , 0 ) y = x2 – 6x + 8 y = x2 – 2x – 24 ( 8 , 24 ) y = x2 – 6x + 8 y = x2 + 10x + 24 (– 1 , 15 ) y = x2 – 6x + 9 y = x2 – 4x + 3 ( 3 , 0 ) y = x2 – 5x – 24 y = x2 – 3x – 28 ( 2 , –30 ) y = x2 – 5x + 4 y = x2 – x ( 1 , 0 ) y = x2 – 5x + 4 y = x2 – 6 ( 2 , –2 ) y = x2 – 4x – 7 y = x2 – 3x – 3 ( –4 , 25 ) y = x2 – 4x + 12 y = x2 ( 3, 9 ) y = x2 – 3x – 70 y = x2 + 16x + 63 (– 7 , 0 ) y = x2 – 3x – 28 y = x2 + 13x + 36 (– 4 , 0 ) y = x2 – 2x – 35 y = x2 + 11x + 30 (– 5 , 0 ) y = x2 – 2x – 8 y = x2 + 2x – 48 ( 10 , 72 ) y = x2 – 2xx – 6 y = x2 + 7x + 3 ( –1 , –3 y = x2 – 2x – 1 y = x2 – x – 1 ( 0 , 1 y = x2 – 2x y = x2 + 10x + 24 ( –2 , 8 ) y = x2 – 2x + 4 y = x2 + 2x + 4 ( 0 , 4 ) y = x2 – 64 y = x2 + 14x + 48 (– 8 , 0 ) y = x2 – 6 y = x2 + x – 10 ( 4 , 10 ) y = x2 y = x2 + x – 2 ( 2 , 4 ) y = x2 y = x2 + 3x – 15 ( 5, 25 y = x2 + 5 y = x2 – x + 5 ( 0 , 5 ) y = x2 + 5 y = x2 + 8x – 11 ( 2 , 9 ) y = x2 + x y = x2 + 21x + 20 (– 1 , 0 ) y = x2 + 2x – 3 y = x2 + 8x – 9 ( 1 , 0 y = x2 + 2x + 8 y = x2 + 3x + 1 ( 7 , 71 ) y = x2 + 4x – 5 y = x2 + 6x – 7 ( 1 , 0 )
Kissing or mutual tangent a parabola with a maximum and one with a minimum
parabola 1 parabola 2 point of tangency tangent y = x2 – a y = –x2 – a ( 0 , –a ) y = –a y = x2 + 2ax + x2 = (x + a)2 y = –x2 + 2ax – a2 = – (x + a)2 ( –a , 0) y = 0 y = x2 + ax = x(x + a) y = –x2 + ax = –x(x – a) ( 0 , 0 y = ax y = x2 – 7x + 12 = (x – 4)(x – 3) y = –x2 + 5x – 6 = –(x – 2)(x – 3) ( 3 , 0 ) y = – (x – 3) y = x2 – 6x + 5 = (x – 5)(x – 1) y = –xy = x2 – 7x + 12 = (x – 4)(x – 3) – 2x + 3 = –(x + 3)(x – 1) (1 , 0) y = –4(x – 1) y = x2 – 6x + 8 = (x – 2)(x – 4) y = –x2 + 2x = –x(x – 2) ( 2 , 0 ) y = –2(x – 2) y = x2 – 5x + 4 = (x – 4)(x – 1) y = –x2 – x + 2 = –(x + 2)(x – 1) ( 1 , 0 ) y = –3(x – 1) y = x2 – 5x + 6 = (x – 3)(x – 2) y = –x2 – 5x + 6 = –(x – 1)(x + 6) ( 0 , 6 ) y = –5x + 6 y = x2 – 5x + 6 = (x – 2)(x – 3) y = –x2 + 3x – 2 = –(x – 2)(x – 1) ( 2 , 0 ) y = –(x – 2) y = x2 – 3x – 10 = (x – 5)(x + 2) y = –x2 – 7x – 12 = –(x + 3)(x + 4) ( –1 , –6 y = –5x + 11 y = x2 – 3x + 2 = (x – 2)(x – 1) y = –x2 + 5x – 6 = –(x – 3)(x – 2) ( 2 , 0 ) y = x – 2 y = x2 – x – 12 = (x – 4)(x + 3) y = –x2 – 9x – 20 = –(x + 4)(x + 5) ( –2 , –6 ) y = –5x – 16 y = x2 – x – 2 = (x – 2)(x + 1) y = –x2 – 5x – 4 = –(x + 4)(x + 1) ( –1 , 0 ) y = –3(x + 1) y = x2 – x = x(x – 1) y = – x2 – 9x – 8 = –(x + 1)(x + 8) (–2 , 6) y = –5x – 4 y = x2 – x = x(x – 1) y = –x2 + 3x – 2 = –(x – 2)(x – 1) ( 1 , 0 ) y = x – 1 y = x2 – 4 = (x – 2)(x + 2) y = –x2 – 8x – 12 = –(x + 6)(x + 2) ( –2 , 0 ) y = –4(x + 2) y = x2 – 4 = (x – 2)(x + 2) y = –x2 + 8x – 12 = –(x – 6)(x – 2) ( 2 , 0 ) y = 4(x – 2) y = x2 – 1 = (x + 1)(x – 1) y = –x2 + 4x – 3 = –(x – 3)(x – 1) ( 1 , 0 ) y = 2(x – 1) y = x2 + x – 2 = (x + 2)(x – 1) y = –x2 + 5x – 4 = –(x – 4)(x – 1) ( 1 , 0 ) y = 3(x – 1) y = x2 + x = x(x + 1) y = –x2 + x = –x(x – 1) ( 0 , 0 ) y = x y = x2 + 3x – 4 = (x + 4)(x – 1) y = –x2 + 7x – 6 = –(x – 6)(x – 1) (1 , 0 ) y = 5(x – 1) y = x2 + 4x + 3 = (x + 3)(x + 1) y = –x2 – 8x – 15 = –(x + 3)(x + 5) ( –3 , 0 ) y = –2(x + 3) y = x2 + 5x + 6 = (x + 2)(x + 3) y = –x2 – 7x – 12 = –(x + 3)(x + 4) ( –3 , 0 ) y = –(x + 3) Kissing or mutual tangent parabolas with different size
If a ≠ 0, c ≠ 0 and x ≠ c then the parabolas impinge
y = a(x - b)2 and y = c(x - b)2 each other ( b , 0 )
while y = 0 is the common tangent is.
parabola 1 parabola 2 point of tangency tangent y = –9x2 – 9x = –9x(x + 1) y = –8x2 – 9x = –x(8x + 9) ( 0 , 0 ) y = 0 y = –8x2 + 8x + 16 = –8(x – 2)(x + 1) y = x2 – 10x + 25 = (x – 5)2 ( 1 , 16 ) y = –8(x – 3) y = –7x2 + 6x + 1 = –(7x + 1)(x – 1) y = –2x2 – 4x + 6 = –2(x + 3)(x – 1) ( 1 , 0 ) y = – 8(x – 1) y = –6x2 + 6 = –6(x –1)(x + 1) y = –4x2 – 4x + 8 = –4(x + 2)(x – 1) ( 1 , 0 ) y = – 12(x – 1) y = –5x2 – 10x = –5xx(x + 2) y = 2x2 – 10x = x(2x –10) ( 0 , 0 ) y = – 10x y = –4x2 – 32x – 64 = –4(x + 4)2 y = –3x2 + 6x + 297 = –3(x – 11)(x + 9) ( –19 , –900 ) y = 120x + 3180 =60(2x + 53 y = –4x2 – 8x – 4 = –4(x + 1)2 y = 5x – 8x – 4 = (5x + 2)(x – 2) ( 0 , –4) y = –4(3x + 2) y = –4x2 – x = –x(4x + 1) y = –x2 – x = –x(x + 1) ( 0 , 0 ) y = – x y = –4x2 + 4x + 8 = –4(x – 2)(x + 1) y = –3x2 + 6x + 9 = –3(x – 3)(x + 1) ( –1 , 0 ) y = – 12(x – 1) y = –4x2 + 28x – 48 = –4(x – 4)(x – 3 y = –2x2 + 50 = –2(x – 5)(x + 5) ( 7 , -48 ) y = – 28x + 148 = – 4(7x – 37) y = –3x2 – 6x = –3x(x + 2) y = –x2 + 2x + 8 = –(x – 4)(x + 2) ( –2 , 0 ) y = 6(x + 2) y = –3x2 + x + 2 = –(3x + 2)(x – 1) y = x2 – 7x + 6 = (x – 6)(x – 1 ( 1 , 0 ) y = – 5(x – 1) y = –3x2 + 3x = –3x(x – 1) y = 2x2 – 7x + 5 = (2x – 5)(x – 1) ( 1 , 0 ) y = –3(x – 1) y = –3x2 + 6x = –3x(x – 2) y = x2 – 10x + 16 = (x – 2)(x – 8) ( 2 , 0 ) y = – 6(x – 2) y = –2x2 + 4x + 6 = –2(x – 3)(x + 1 y = –x2 + 6x + 7 = –(x – 7)(x + 1) ( –1 , 0 ) y = 8x + 8 y = –2x2 + 8x = –2x(x – 4) y = 4x2 + 8x = 4x(x + 2) ( 0 , 0 ) y = 8x y = –x2 – 4x + 5 = –(x + 5)(x – 1) y = 3x2 – 12x + 9 = 3(x – 3)(x – 1) ( 1 , 0 ) y = – 6x + 6 y = x2 – x = x(x – 1) y = 3x2 – x = x(3x – 1) ( 0 , 0 ) y = – x y = x2 – 4 = (x – 2)(x + 2) y = 2x2 – 4x = 2x(x – 2) ( 2 , 0 ) y = 4(x – 2) y = x2 + 2x – 8 = (x – 2)(x + 4) y = 2x2 – 2x – 4 = 2(x – 2)(x + 1) ( 2 , 0 ) y = 6(x – 2) y = x2 + 4x + 3 = (x + 3)(x + 1) y = –2x2 – 2x = –2x(x + 1) ( –1 , 0 ) y = 2(x + 1) y = 3x2 – 4x – 4 = (3x + 2)(x – 2) y = 4x2 ( –2 , 0 ) y = – 16(x + 1)
Cubic curve with a horizontal bend tangent
The cubic curve f(x) = ax3 + bx2 + {b2/(3a)}x has a horizontal bend tangent
at the bending point ( –b/(3a) , {b/(3a)}3)
The cubic curve bend point f(x) = –3x3 + 9x2 – 9x ( 1 , –3 ) f(x) = –2x3 – 6x2 – 6x ( –1 , 2) f(x) = – x3 + 3x2 – 3x ( 1 , –1 ) f(x) = x3 + 3x2 + 3x ( -1 , –1 ) f(x) = 2x3 + 6x2 + 6x ( –1 , –2 ) f(x) = 3x3 + 9x2 + 9x ( –1 , –3 ) f(x) = 4x3 + 24x2 + 48x ( –2 , –32 )
The orthogonal hyperbola
The hyperbola y=(ax + b)/(x + c) intersects its mirror axis y = x + a + c in
S1 = ( –c+√(b–ac) , a +√(b–ac)) and S2 = (–c – √(b–ac) , a – √(b–ac)).
The center of the hyperbola is C = ( –c, a ).
a b c S1 S2 1 3 2 ( –1 , 2 ) ( –3 , 0) 1 5 1 ( 1 , 3 ) ( –3 , –1 ) 1 5 4 ( –3 , 2 ) ( –5 , 0 ) 1 10 1 ( 2 , 4 ) ( –4 , –2 ) 1 11 2 ( 1 , 4 ) ( –5 , –2 ) 2 6 1 ( 1 , 4 ) ( –3 , 0) 2 7 3 ( –2 , 3 ) ( –4 , 1) 2 8 2 ( 0 , 4 ) ( –4 , 0 ) 2 11 1 ( 2 , 5 ) ( –4 , –1 ) 3 7 2 ( –1 , 4 ) (–3 , 2) 4 5 1 ( 0 , 5 ) ( –2 , 3)
Math Sciences