What is the minimum vertical distance between the parabolas y = x² + 1 and y = x - x²

Solution:

Given two parabolas,

y = x² + 1 ------>(1)

and y = x - x²  ----->(2)

The vertical distance 'd' between the parabolas is determined by their axes of symmetry.

axes of symmetry of curve (1) is x = 0

axes of symmetry of curve (2) is x = ½

To find the midpoint of both the axes of symmetry, we use the midpoint formula.

X = (0+½)/2

⇒ X = ¼

Key-in X = ¼ in (1)

⇒ Y = 1/16 + 1

Y = 17/16

⇒ Point A = (1/4,17/16)

Substituting X = ¼ in (2)

⇒Y = - 1/16 + ¼ 

Y = 3/16

⇒ Point B = (1/4, 3/16)

AB = 17/16 - 3/16

=14/16

AB = 7/8 units.

Therefore, the minimum vertical distance between the curves is 7/8 units.

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What is the minimum vertical distance between the parabolas y = x² + 1 and y = x - x²

Summary:

The minimum vertical distance between the parabolas y = x² + 1 and y = x - x² is 7/8 units.