## 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 =x^{2}+ 5x+ 3

y= 3x+ 3 intersectsy=x^{2}+ 5x+ 3 in (–2 , –3) and (0 , 3)

y= 3x+ 6 intersectsy=x^{2}+ 5x+ 3 in (–3 , –3) and (1 , 9)

y= 3x+ 11 intersectsy=x^{2}+ 5x+ 3 in (–4 , –1) and (2 , 17)

y= 3x+ 18 intersectsy=x^{2}+ 5x+ 3 in (–5 , 3) and (3 , 27)

y= –3x– 13 is in ( –4 , –1) the tangent aty=x^{2}+ 5x+ 3

y= –3x– 12 intersectsy=x^{2}+ 5x+ 3 in (-5 , 3) and (-3 , -3)

y= –3x– 9 intersectsy=x^{2}+ 5x+ 3 in (-6 , 9) and (-2 , -3)

y= –3x– 4 intersectsy=x^{2}+ 5x+ 3 in (-7 , 17) and (-1 , -1)

y= –3x+ 3 intersectsy=x^{2}+ 5x+ 3 in (-8 , 27) and ( 0 , 3)

y= 3x– 13 is in (4 , -1) the tangent aty=x^{2}– 5x+ 3

y= 3x– 12 intersectsy=x^{2}– 5x+ 3 in ( 3 , –3) and ( 5 , 3)

y= 3x– 9 intersectsy=x^{2}– 5x+ 3 in ( 2 , –3) and ( 6 , 9)

y= 3x+ 3 intersectsy=x^{2}– 5x+ 3 in ( 0 , 3) and ( 8 , 27)

y= 3x+ 12 intersectsy=x^{2}– 5x+ 3 in (–1 , 9) and ( 9 , 39)

y= 3x+ 23 intersectsy=x^{2}– 5x+ 3 in (–2 , 17) and (10 , 53)

y= –3x– 19 is in (4 , –7 ) the tangent aty=x^{2}– 5x+ 3

y= –3x–18 intersectsy=x^{2}– 5x+ 3 in (3 , –9) and (5 , –3)

y= –3x–10 intersectsy=x^{2}– 5x+ 3 in (1 , –7) and (7 , 11)

y= –3x– 3 intersectsy=x^{2}– 5x+ 3 in (0 , –3) and (8 , 21)

y= –3x– 6 intersectsy=x^{2}– 5x+ 3 in (–1 , 3) and (9 , 33)

y= 3x– 4 is in (1 , -1) the tangent aty=x^{2}+x– 3

y= 3x– 3 intersects the parabola in ( 0 , –3) and (2 , 3)

y= 3xintersects 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 aty=x^{2}+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 aty=x^{2}– 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 aty=x^{2}+ 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 parabolay=x^{2}– 4x+ 3 = (x– 1)(x– 3)

In (–3 , 24) the tangent isy= –10x– 6

In (–2 , 15) the tangent isy= – 8xx– 1

In (–1 , 8) the tangent isy= – 6x+ 2

In ( 0 , 3) the tangent isy= – 4x+ 3

In ( 1 , 0) the tangent isy= – 2x+ 2

In ( 2 , –1) the tangent isy= – 1

In ( 3 , 0) the tangent isy= 2x– 6

In ( 4 , 3) the tangent isy= 4x– 13

In ( 5 , 8) the tangent isy= 6x– 22

In ( 6 , 15) the tangent isy= 8x– 33

In ( 7 , 24) the tangent isy= 10x– 46

Choose the parabolay=x^{2}– 4x– 21 = (x+ 3)(xx– 7)

In (–3 , 0) the tangent isy= –10x– 30

In (–2 , –9) the tangent isy= – 8x – 25

In (–1 , –16) the tangent isy= – 6x – 22

In ( 0 , –21) the tangent isy= – 4x – 21

In ( 1 , –24) the tangent isy= – 2x – 22

In ( 2 , –25) the tangent isy= – 25

In ( 3 , –24) the tangent isy= 2x – 30

In ( 4 , –21) the tangent isy= 4x – 37

In ( 5 , –16) the tangent isy= 6x – 46

In ( 6 , – 9) the tangent isy= 8x – 57

In ( 7 , 0) the tangent isy= 10x – 70

Parabolas with one intersection

Looking for the intersection the square term cancels.

parabola 1 parabola 2 intersect at y=x^{2}– 16x+ 60y=x^{2}– 8x+ 12x( 6 , 0 ) y=x^{2}– 14x+ 40y=x^{2}+ 2x– 8( 3 , 7 ) y=x^{2}– 14x+ 56y=x^{2}– 7x( 8 , 8 ) y=x^{2}– 13x+ 20y=x^{2}– 10x+ 29( –3 , 68 ) y=x^{2}– 13x+ 30y=x^{2}–x– 42( 6 , – 12 ) y=x^{2}– 13x+ 42y=x^{2}+x( 3 , 12 ) y=x^{2}– 12x+ 35y=x^{2}– 7x+ 10( 5 , 0 y=x^{2}– 12x+ 42y=x^{2}– 6x( 7 , 7 ) y=x^{2}– 11x+ 10y=x^{2}– 5x+ 4( 1 , 0 ) y=x^{2}– 10x+ 21y=x^{2}– 9x+ 18( 3 , 0 ) y=x^{2}– 1x+ 22y=x^{2}– 9x+ 21( 1 , 13 ) y=x^{2}– 9x+ 7y=x^{2}– 5x– 1( 2 , –7 ) y=x^{2}– 8x+ 22y=x^{2}– 5x+ 19( 1 , 15 ) y=x^{2}– 7x+ 6y=x^{2}– 5x( 3 , –6 ) y=x^{2}–x6xy=x^{2}–x– 20( 4 , –8 ) y=x^{2}– 6x+ 8y=x^{2}+x– 6( 2 , 0 ) y=x^{2}– 6x+ 8y=x^{2}– 2x– 24( 8 , 24 ) y=x^{2}– 6x+ 8y=x^{2}+ 10x+ 24(– 1 , 15 ) y=x^{2}– 6x+ 9y=x^{2}– 4x+ 3( 3 , 0 ) y=x^{2}– 5x– 24y=x^{2}– 3x– 28( 2 , –30 ) y=x^{2}– 5x+ 4y=x^{2}–x( 1 , 0 ) y=x^{2}– 5x+ 4y=x^{2}– 6( 2 , –2 ) y=x^{2}– 4x– 7y=x^{2}– 3x– 3( –4 , 25 ) y=x^{2}– 4x+ 12y=x^{2}( 3, 9 ) y=x^{2}– 3x– 70y=x^{2}+ 16x+ 63(– 7 , 0 ) y=x^{2}– 3x– 28y=x^{2}+ 13x+ 36(– 4 , 0 ) y=x^{2}– 2x– 35y=x^{2}+ 11x+ 30(– 5 , 0 ) y=x^{2}– 2x– 8y=x^{2}+ 2x– 48( 10 , 72 ) y=x^{2}– 2xx– 6y=x^{2}+ 7x+ 3( –1 , –3 y=x^{2}– 2x– 1y=x^{2}–x– 1( 0 , 1 y=x^{2}– 2xy=x^{2}+ 10x+ 24( –2 , 8 ) y=x^{2}– 2x+ 4y=x^{2}+ 2x+ 4( 0 , 4 ) y=x^{2}– 64y=x^{2}+ 14x+ 48(– 8 , 0 ) y=x^{2}– 6y=x^{2}+x– 10( 4 , 10 ) y=x^{2}y=x^{2}+x– 2( 2 , 4 ) y=x^{2}y=x^{2}+ 3x– 15( 5, 25 y=x^{2}+ 5y=x^{2}–x+ 5( 0 , 5 ) y=x^{2}+ 5y=x^{2}+ 8x– 11( 2 , 9 ) y=x^{2}+xy=x^{2}+ 21x+ 20(– 1 , 0 ) y=x^{2}+ 2x– 3y=x^{2}+ 8x– 9( 1 , 0 y=x^{2}+ 2x+ 8y=x^{2}+ 3x+ 1( 7 , 71 ) y=x^{2}+ 4x– 5y=x^{2}+ 6x– 7( 1 , 0 ) Go up

Kissing or mutual tangent a parabola with a maximum and one with a minimum

parabola 1 parabola 2 point of tangency tangent y=x^{2}–ay= –x^{2}–a( 0 , –a)y = –ay=x^{2}+ 2ax+x2 = (x + a)^{2}y= –x2 + 2ax – a2 = – (x + a)^{2}( – a, 0)y= 0y=x^{2}+ax = x(x + a)y= –x^{2}+ax = –x(x – a)( 0 , 0 y = axy = x^{2}– 7x+ 12 = (x– 4)(x– 3)y = –x^{2}+ 5x– 6 = –(x– 2)(x– 3)( 3 , 0 ) y= – (x– 3)y = x^{2}– 6x+ 5 = (x– 5)(x– 1)y = –xy = x^{2}– 7x+ 12 = (x– 4)(x– 3) – 2x+ 3 = –(x+ 3)(x– 1)(1 , 0) y= –4(x– 1)y = x^{2}– 6x+ 8 = (x– 2)(x– 4)y = –x^{2}+ 2x = –x(x– 2)( 2 , 0 ) y= –2(x– 2)y = x^{2}– 5x+ 4 = (x– 4)(x– 1)y = –x^{2}–x+ 2 = –(x+ 2)(x– 1)( 1 , 0 ) y= –3(x– 1)y = x^{2}– 5x+ 6 = (x– 3)(x– 2)y = –x^{2}– 5x+ 6 = –(x– 1)(x+ 6)( 0 , 6 ) y= –5x+ 6y = x^{2}– 5x+ 6 = (x– 2)(x– 3)y = –x^{2}+ 3x– 2 = –(x– 2)(x– 1)( 2 , 0 ) y= –(x– 2)y = x^{2}– 3x– 10 = (x– 5)(x+ 2)y = –x^{2}– 7x– 12 = –(x+ 3)(x+ 4)( –1 , –6 y= –5x+ 11y = x^{2}– 3x+ 2 = (x– 2)(x– 1)y = –x^{2}+ 5x– 6 = –(x– 3)(x– 2)( 2 , 0 ) y = x– 2y = x^{2}–x– 12 = (x– 4)(x+ 3)y = –x^{2}– 9x– 20 = –(x+ 4)(x+ 5)( –2 , –6 ) y= –5x– 16y = x^{2}–x– 2 = (x– 2)(x+ 1)y = –x^{2}– 5x– 4 = –(x+ 4)(x+ 1)( –1 , 0 ) y= –3(x+ 1)y = x^{2}–x = x(x– 1)y = – x^{2}– 9x– 8 = –(x+ 1)(x+ 8)(–2 , 6) y= –5x– 4y = x^{2}–x = x(x– 1)y = –x^{2}+ 3x– 2 = –(x– 2)(x– 1)( 1 , 0 ) y = x– 1y = x^{2}– 4 = (x– 2)(x+ 2)y = –x^{2}– 8x– 12 = –(x+ 6)(x+ 2)( –2 , 0 ) y= –4(x+ 2)y = x^{2}– 4 = (x– 2)(x+ 2)y = –x^{2}+ 8x– 12 = –(x– 6)(x– 2)( 2 , 0 ) y= 4(x– 2)y = x^{2}– 1 = (x+ 1)(x– 1)y = –x^{2}+ 4x– 3 = –(x– 3)(x– 1)( 1 , 0 ) y= 2(x– 1)y = x^{2}+x– 2 = (x+ 2)(x– 1)y = –x2 + 5x– 4 = –(x– 4)(x– 1)( 1 , 0 ) y= 3(x– 1)y = x^{2}+x = x(x+ 1)y = –x^{2}+x= –x(x– 1)( 0 , 0 ) y = xy = x^{2}+ 3x– 4 = (x+ 4)(x– 1)y = –x^{2}+ 7x– 6 = –(x– 6)(x– 1)(1 , 0 ) y= 5(x– 1)y = x^{2}+ 4x+ 3 = (x+ 3)(x+ 1)y = –x^{2}– 8x– 15 = –(x+ 3)(x+ 5)( –3 , 0 ) y= –2(x+ 3)y = x^{2}+ 5x+ 6 = (x+ 2)(x+ 3)y = –x^{2}– 7x– 12 = –(x+ 3)(x+ 4)( –3 , 0 ) y= –(x+ 3)Kissing or mutual tangent parabolas with different size

Ifa≠ 0,c≠ 0 andx ≠ cthen the parabolas impinge

y = a(x - b)^{2}andy = c(x - b)^{2}each other (b, 0 )

whiley= 0 is the common tangent is.

parabola 1 parabola 2 point of tangency tangent y= –9x^{2}– 9x= –9x(x+ 1)y= –8x^{2}– 9x= –x(8x+ 9)( 0 , 0 ) y= 0y= –8x^{2}+ 8x+ 16 = –8(x– 2)(x+ 1)y = x^{2}– 10x+ 25 = (x– 5)^{2}( 1 , 16 ) y= –8(x– 3)y= –7x^{2}+ 6x+ 1 = –(7x+ 1)(x– 1)y= –2x^{2}– 4x+ 6 = –2(x+ 3)(x– 1)( 1 , 0 ) y= – 8(x– 1)y= –6x^{2}+ 6 = –6(x–1)(x+ 1)y= –4x^{2}– 4x+ 8 = –4(x+ 2)(x– 1)( 1 , 0 ) y= – 12(x– 1)y= –5x^{2}– 10x= –5xx(x+ 2)y= 2x^{2}– 10x=x(2x–10)( 0 , 0 ) y= – 10xy= –4x^{2}– 32x– 64 = –4(x+ 4)^{2}y= –3x^{2}+ 6x+ 297 = –3(x– 11)(x+ 9)( –19 , –900 ) y= 120x+ 3180 =60(2x+ 53y= –4x^{2}– 8x– 4 = –4(x+ 1)^{2}y= 5x– 8x– 4 = (5x+ 2)(x– 2)( 0 , –4) y= –4(3x + 2)y= –4x^{2}–x= –x(4x+ 1)y = –x^{2}–x= –x(x+ 1)( 0 , 0 ) y = – xy= –4x^{2}+ 4x+ 8 = –4(x– 2)(x+ 1)y= –3x^{2}+ 6x+ 9 = –3(x– 3)(x+ 1)( –1 , 0 ) y= – 12(x– 1)y= –4x^{2}+ 28x– 48 = –4(x– 4)(x– 3y= –2x^{2}+ 50 = –2(x– 5)(x+ 5)( 7 , -48 ) y= – 28x+ 148 = – 4(7x– 37)y= –3x^{2}– 6x= –3x(x+ 2)y = –x^{2}+ 2x+ 8 = –(x– 4)(x+ 2)( –2 , 0 ) y= 6(x+ 2)y= –3x^{2}+x+ 2 = –(3x+ 2)(x– 1)y = x^{2}– 7x+ 6 = (x– 6)(x– 1( 1 , 0 ) y= – 5(x– 1)y= –3x^{2}+ 3x= –3x(x– 1)y= 2x^{2}– 7x+ 5 = (2x– 5)(x– 1)( 1 , 0 ) y= –3(x– 1)y= –3x^{2}+ 6x= –3x(x– 2)y = x^{2}– 10x+ 16 = (x– 2)(x– 8)( 2 , 0 ) y= – 6(x– 2)y= –2x^{2}+ 4x+ 6 = –2(x– 3)(x+ 1y = –x^{2}+ 6x+ 7 = –(x– 7)(x+ 1)( –1 , 0 ) y= 8x+ 8y= –2x^{2}+ 8x= –2x(x– 4)y= 4x^{2}+ 8x= 4x(x+ 2)( 0 , 0 ) y= 8xy = –x^{2}– 4x+ 5 = –(x+ 5)(x– 1)y= 3x^{2}– 12x+ 9 = 3(x– 3)(x– 1)( 1 , 0 ) y= – 6x+ 6y = x^{2}–x = x(x– 1)y= 3x^{2}–x = x(3x– 1)( 0 , 0 ) y = – xy = x^{2}– 4 = (x– 2)(x+ 2)y= 2x^{2}– 4x= 2x(x– 2)( 2 , 0 ) y= 4(x– 2)y = x^{2}+ 2x– 8 = (x– 2)(x+ 4)y= 2x^{2}– 2x– 4 = 2(x– 2)(x+ 1)( 2 , 0 ) y= 6(x– 2)y = x^{2}+ 4x+ 3 = (x+ 3)(x+ 1)y= –2x^{2}– 2x= –2x(x+ 1)( –1 , 0 ) y= 2(x+ 1)y= 3x^{2}– 4x– 4 = (3x+ 2)(x– 2)y= 4x^{2}( –2 , 0 ) y= – 16(x+ 1)

## Cubic curve with a horizontal bend tangent

The cubic curve f(x) =ax^{3}+bx^{2}+ {b^{2}/(3a)}x has a horizontal bend tangent

at the bending point ( –b/(3a) , {b/(3a)}^{3})

The cubic curve bend point f( x) = –3x^{3}+ 9x^{2}– 9x( 1 , –3 ) f( x) = –2x^{3}– 6x^{2}– 6x( –1 , 2) f( x) = –x^{3}+ 3x^{2}– 3x( 1 , –1 ) f( x) =x^{3}+ 3x^{2}+ 3x( -1 , –1 ) f( x) = 2x^{3}+ 6x^{2}+ 6x( –1 , –2 ) f( x) = 3x^{3}+ 9x^{2}+ 9x( –1 , –3 ) f( x) = 4x^{3}+ 24x^{2}+ 48x( –2 , –32 ) ## The orthogonal hyperbola

The hyperbolay=(ax + b)/(x + c) intersects its mirror axisy = x + a + cin

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 cS1 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)

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