Parallel Tangents to Cubic Graph

In summary, the conversation discusses finding the equation of a tangent line on the graph of the function f(x)=(x+1)(x+4)(x+6) that is parallel to another tangent line with an equation of y=10x+60. The speaker is looking for assistance in finding the equation algebraically, as they have already found a point where the tangent is parallel using a computer program. They know that the tangent they are looking for will have a slope of 10.
  • #1
flyinghigh
9
0

Homework Statement


I have the graph of a function f(x)=(x+1)(x+4)(x+6). I've found the tangent at x=-6, the equation of which is y=10x+60. I then need to algebraically find the equation of another tangent on the curve which is parallel to the first.

Homework Equations


No idea.

The Attempt at a Solution


Since I'm graphing this on Autograph I've managed to find a point where the tangent is parallel to the first, but since this can be done without the computer program I'm really interested to know how it's done. So yeah, I know that the tangent I'm trying to find will have a gradient of 10 but that's about it...

Any help would be great,
flyinghigh
 
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  • #2
Okay, you know you're looking for a tangent line with a slope of 10.

Can you write an equation for the slope of the tangent (as a function of x), and set that equal to 10?
 

FAQ: Parallel Tangents to Cubic Graph

1. What is a parallel tangent to a cubic graph?

A parallel tangent to a cubic graph is a line that touches the graph at two points and has the same slope as the graph at those points. This means that the line is never above or below the graph, but rather "runs alongside" it.

2. How do you find the equation of a parallel tangent to a cubic graph?

To find the equation of a parallel tangent to a cubic graph, you need to first find the slope of the tangent line at the given points of tangency. This can be done by finding the derivative of the cubic function. Then, using the slope and the given points, you can use the point-slope form of a line to write the equation of the parallel tangent.

3. Can a cubic graph have multiple parallel tangents?

Yes, a cubic graph can have multiple parallel tangents. This can happen when the graph has multiple points where the slope is the same. In this case, there will be multiple lines that are parallel to the graph at those points of tangency.

4. How does the degree of a cubic graph affect the number of parallel tangents it can have?

The degree of a cubic graph does not have a direct impact on the number of parallel tangents it can have. However, a higher degree cubic graph may have more points of tangency, which could potentially result in more parallel tangents.

5. Are parallel tangents to a cubic graph always horizontal?

No, parallel tangents to a cubic graph are not always horizontal. They can be any slope that is equal to the slope of the cubic graph at the points of tangency. This means that the parallel tangent can be horizontal, vertical, or any other slope in between.

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