Vertical distance between two parabola

In summary, the conversation is discussing finding the vertical distance between two parabolas and the minimum value for that distance. The equation for the distance is d(x) = y1 - y2, and the minimum value was found to be 2. The conversation then moves on to discussing finding the value of k that will result in the graphs of y = kx and y = x^2 + 25 intersecting at only one point. The solution involves using the discriminant method and setting the equations equal to find the value of k. The conversation ends with a clarification on the variables used in the equations.
  • #1
icosane
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0

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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  • #2
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?
 
  • #3
Yes, I think it does.
 
  • #4
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
 
  • #5
oh i forgot...

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

tnx tnx ^__^__^
 

1. What is the formula for finding the vertical distance between two parabolas?

The formula for finding the vertical distance between two parabolas is:
d = |h1 - h2|
Where d is the vertical distance, h1 is the y-coordinate of the vertex of the first parabola, and h2 is the y-coordinate of the vertex of the second parabola.

2. How do you find the vertices of two parabolas?

To find the vertices of two parabolas, you need to equate the equations of the parabolas to each other and solve for the x-value. This x-value will be the x-coordinate of the vertices. To find the y-coordinate, plug the x-value into either of the equations and solve for y.

3. Can the vertical distance between two parabolas be negative?

Yes, the vertical distance between two parabolas can be negative. This indicates that one parabola is below the other parabola.

4. How does changing the coefficients of a parabola affect the vertical distance between two parabolas?

Changing the coefficients of a parabola can affect the vertical distance between two parabolas in several ways. If the coefficient of x2 is increased, the parabola becomes narrower and the vertical distance between two parabolas may decrease. If the coefficient of x2 is decreased, the parabola becomes wider and the vertical distance may increase. Additionally, changing the coefficients of x and the constant term can also affect the vertical distance between two parabolas.

5. Is the vertical distance between two parabolas always constant?

No, the vertical distance between two parabolas is not always constant. It can vary depending on the equations of the parabolas and their respective vertices. However, if the equations of the parabolas are identical, the vertical distance between them will always be zero.

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