Vertical distance between two parabola

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Homework Help Overview

The discussion revolves around finding the vertical distance between two parabolas defined by the equations y = x^2 + 6 and y = -(x-2)^2 + 6. The original poster seeks to establish a function for this distance and determine its minimum value.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster initially presents a function for the vertical distance and refers to a solution found in a textbook. They later express a realization about the simplicity of the distance calculation. Another participant suggests using the discriminant method to find conditions for a single intersection point between a linear function and a parabola.

Discussion Status

Participants are exploring different aspects of the problems presented. Some guidance has been offered regarding the use of the discriminant method to determine the conditions for intersections, but no consensus has been reached on the specific values of k or the minimum distance.

Contextual Notes

The original poster mentions self-study and the absence of a teacher for assistance, indicating a potential constraint in seeking clarification on the concepts discussed.

icosane
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Homework Statement



Write a function for d(x), the vertical distance between the two curves, and find the minimum value of d(x).

Homework Equations



The equation for parabola one is y = x^2 + 6, for parabola two, y = -(x-2)^2 + 6


The Attempt at a Solution



The answer in the back of the book is d(x) = 2x^2 - 4x + 4, with a minimum value of 2. This is from my old algebra 2 trig textbook and I have no teacher to ask for help, as I am doing self study. Any and all help would be very much appreciated.
 
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I figured it out, its just d(x) = y1 - y2, seems I was just over thinking things... but I have another question :)

Find the Value(s) of k for which the graph y = kx intersects the graph of y = x^2 + 25 in only one point.

If I set the equations equal, I get 0 = x^2 - kx + 25, and I know perfect squares have only one root, so it seems as though the answer is 10 or -10. Does this seem reasonable?
 
Yes, I think it does.
 
from 0 = x2 - kx + 25, use the discriminant method to find the value of k that will result to a single intercept only. that is

b2 - 4ac = 0 for one root or one intercept
b2 - 4ac > 0 for two roots or two intercepts
b2 - 4ac < 0 if you wish them to have no intercepts
 
oh i forgot...

a = 1
b = -k
c = 25
x2 = x squared

tnx tnx ^__^__^
 

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