Finding Horizontal Tangent Lines on a Parabola

The problem involves a graph of the function y = x2 - 4x + 5 and requires using the concept of completing the square to find the vertex and coordinates. The solution involves finding the square roots of the y-values on the parabola. In summary, Mark needs help with finding values of x=c for a problem involving a tangent line to a graph and the solution involves using the concept of completing the square.
  • #1
AFNequation
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Homework Statement



I have a homework that i couldn't do :( can you explain it to me please ?
the problem is :
find all values of x=c so that the tangent line to the graph of f(x) af (c , f(c)) will be horizontal
http://img13.imageshack.us/img13/4222/scan0002izs.jpg





The Attempt at a Solution


i know in the end the answer will be x = 2 ... but i want to how did we get it
 
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  • #2
Consider y = x2 - 4x + 5.

The graph of this equation is a parabola that opens up. By completing the square you can find the vertex and the coordinates of the vertex. The tangent line to this graph is horizontal at the vertex.

The function you have is related to this in that all of its y-values are the square roots of the y-values on the parabola.

Hope this is enough of a hint.
Mark
 

FAQ: Finding Horizontal Tangent Lines on a Parabola

1. What is a tangent line?

A tangent line is a straight line that touches a curve at exactly one point. It represents the slope of the curve at that specific point.

2. How do you find the equation of a tangent line?

To find the equation of a tangent line, you need to first find the slope of the curve at the point of tangency. This can be done by taking the derivative of the curve at that point. Then, using the point-slope form of a line, you can plug in the slope and the coordinates of the point of tangency to find the equation of the tangent line.

3. What does it mean for a tangent line to be horizontal?

A tangent line is horizontal when its slope is equal to 0. This means that the curve is not changing at that point and is instead flat or level.

4. How can you determine if a curve has a horizontal tangent line?

A curve has a horizontal tangent line at a specific point if the derivative of the curve at that point is equal to 0. This means that the slope of the curve at that point is 0, making the tangent line horizontal.

5. Why are horizontal tangent lines important in calculus?

Horizontal tangent lines are important in calculus because they can help us find critical points, or points where the derivative is equal to 0. These points can give us valuable information about the behavior of a function and can be used to find maximum and minimum values, as well as points of inflection.

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