Determine whether lines bt and mv are parallel, perpendicular, or neither. you must show all of your work to earn full credit. b(1,-4), t(5,12), m(-8,3), v(-4,2)

BT ⊥ MV

The direction vector BT is ...

... T - B = (5, 12) - (1, -4) = (4, 16)

The direction vector MV is ...

... V - M = (-4, 2) - (-8, 3) = (4, -1)

One is not a multiple of the other, so the lines are not parallel.

The dot-product of these direction vectors is

... (4, 16)•(4, -1) = 4·4 + 16·(-1) = 0

When the dot-product is zero, the vectors are perpendicular.

the lines are perpendicular

to determine which case is true we require the slope m of the lines

Parallel lines have equal slopes

Perpendicular slopes are the negative inverse of each other

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = B(1, - 4 ) and (x₂, y₂ ) = T(5, 12 )

= = = 4

repeat with

(x₁, y₁ ) = M(-8, 3 ) and (x₂, y₂ ) = V(-4, 2 )

= = -

4 and - are negative inverses, hence

BT and MV are perpendicular to each other

step-by-step explanation:

here's what happens:

1) the graph of f   = x^2 is shifted 2 units to the right.

2) the resulting graph is reflected in the x-axis, so that the parabola now opens down.

3) this final result is transformed up by 5 units.