Integer intersections and tangents

Intersection lines with parabolas 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)

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.
parabola 1parabola 2intersect at
y = x2 – 16x + 60y = x2 – 8x + 12x( 6 , 0 )
y = x2 – 14x + 40y = x2 + 2x – 8( 3 , 7 )
y = x2 – 14x + 56y = x2 – 7x( 8 , 8 )
y = x2 – 13x + 20y = x2 – 10x + 29( –3 , 68 )
y = x2 – 13x + 30y = x2x – 42( 6 , – 12 )
y = x2 – 13x + 42y = x2 + x( 3 , 12 )
y = x2 – 12x + 35y = x2 – 7x + 10( 5 , 0
y = x2 – 12x + 42y = x2 – 6x( 7 , 7 )
y = x2 – 11x + 10y = x2 – 5x + 4( 1 , 0 )
y = x2 – 10x + 21y = x2 – 9x + 18( 3 , 0 )
y = x2 – 1x + 22y = x2 – 9x + 21( 1 , 13 )
y = x2 – 9x + 7y = x2 – 5x – 1( 2 , –7 )
y = x2 – 8x + 22y = x2 – 5x + 19( 1 , 15 )
y = x2 – 7x + 6y = x2 – 5x( 3 , –6 )
y = x2x6xy = x2x – 20( 4 , –8 )
y = x2 – 6x + 8y = x2 + x – 6( 2 , 0 )
y = x2 – 6x + 8y = x2 – 2x – 24( 8 , 24 )
y = x2 – 6x + 8y = x2 + 10x + 24(– 1 , 15 )
y = x2 – 6x + 9y = x2 – 4x + 3 ( 3 , 0 )
y = x2 – 5x – 24y = x2 – 3x – 28( 2 , –30 )
y = x2 – 5x + 4y = x2x( 1 , 0 )
y = x2 – 5x + 4y = x2 – 6( 2 , –2 )
y = x2 – 4x – 7y = x2 – 3x – 3( –4 , 25 )
y = x2 – 4x + 12y = x2( 3, 9 )
y = x2 – 3x – 70y = x2 + 16x + 63(– 7 , 0 )
y = x2 – 3x – 28y = x2 + 13x + 36(– 4 , 0 )
y = x2 – 2x – 35y = x2 + 11x + 30(– 5 , 0 )
y = x2 – 2x – 8y = x2 + 2x – 48( 10 , 72 )
y = x2 – 2xx – 6y = x2 + 7x + 3( –1 , –3
y = x2 – 2x – 1y = x2x – 1( 0 , 1
y = x2 – 2xy = x2 + 10x + 24( –2 , 8 )
y = x2 – 2x + 4y = x2 + 2x + 4( 0 , 4 )
y = x2 – 64y = x2 + 14x + 48(– 8 , 0 )
y = x2 – 6y = x2 + x – 10( 4 , 10 )
y = x2y = x2 + x – 2( 2 , 4 )
y = x2y = x2 + 3x – 15( 5, 25
y = x2 + 5y = x2x + 5( 0 , 5 )
y = x2 + 5y = x2 + 8x – 11( 2 , 9 )
y = x2 + xy = x2 + 21x + 20(– 1 , 0 )
y = x2 + 2x – 3y = x2 + 8x – 9( 1 , 0
y = x2 + 2x + 8y = x2 + 3x + 1( 7 , 71 )
y = x2 + 4x – 5y = x2 + 6x – 7( 1 , 0 )

Impinging a parabola with a maximum and one with a minimum

parabola 1parabola 2point of tangencytangent
y = x2ay = –x2a( 0 , –a )y = –a
y = x2 + 2ax + x2 = (x + a)2y = –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 = –x2x + 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 = x2x – 12 = (x – 4)(x + 3)y = –x2 – 9x – 20 = –(x + 4)(x + 5)( –2 , –6 )y = –5x – 16
y = x2x – 2 = (x – 2)(x + 1)y = –x2 – 5x – 4 = –(x + 4)(x + 1)( –1 , 0 )y = –3(x + 1)
y = x2x = x(x – 1)y = – x2 – 9x – 8 = –(x + 1)(x + 8)(–2 , 6)y = –5x – 4
y = x2x = 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)

Impinging 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 1parabola 2point of tangencytangent
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)2y = –3x2 + 6x + 297 = –3(x – 11)(x + 9)( –19 , –900 )y = 120x + 3180 =60(2x + 53
y = –4x2 – 8x – 4 = –4(x + 1)2y = 5x – 8x – 4 = (5x + 2)(x – 2)( 0 , –4)y = –4(3x + 2)
y = –4x2x = –x(4x + 1)y = –x2x = –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 – 3y = –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 + 1y = –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 = x2x = x(x – 1)y = 3x2x = 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 curvebend 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 cS1S2
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