Finding Tangent Lines for Parabolas: A Mathematical Approach

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In summary, the conversation discusses finding two lines whose product creates a parabola and are tangent to it. The first step is to realize that the parabola's equation is a product of two lines, and the lines must intersect at the parabola's zeroes. To find the appropriate slope at the zeroes, the derivative is taken and a simple condition is needed. However, the person is stuck as they do not know how to differentiate or have a formula for finding tangent lines. The conversation ends with the realization that calculus is not necessary and the simpler approach is to find how many times the lines intersect.
  • #1
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Ok, I need to find 2 lines that when multiplied (y=x*y=x yields y=x^2)
create a parabola in which those two lines are tangent to it. My problem is that I have no idea where to start. I think that the two lines have to have opposite slopes, or at least one slope must be positive and one must be negative, else both lines would follow the same direction, and there is an obvious problem there...
 
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  • #2
This was surprisingly tricky. The first thing to realize is that if a parabola can be expressed as the product of two lines ax+b and cx+d, then the parabola's equation is (ax+b)(cx+d) and the lines factor it. The next thing to realize is that the parabola must touch a line at each zero of the parabola, and if the line is not tangent at the zero it is not tangent anywhere. Now differentiate (ax+b)(cx+d) (it's easier later on if you do it using the product rule rather than expanding it out first). Then, you must find a simple condition to make the slope of the parabola at the zeroes appropriate, and then solve that condition.

There is also the trivial case of y1 = 0 = y2 but since you ask for a parabola I don't think that is a valid answer.

Edit: it's interesting, the two lines must always intersect at the same y coordinate, 0.5.

What class is this for?
 
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  • #3
It's for my Honors Algebra II class. Not everyone has to do it, my teacher and I are working together on upper level math like this. I can generally get the stuff we discuss, but some of it goes right over my head, as I'm only in 9th grade. But thanks a lot for the help on this one, it's greatly appriceated.
 
  • #4
Well, if you're only in algebra II I don't see how you can do it without differentiating. Are you given a formula to tell if a line is tangent to a parabola?
 
  • #5
Nope, and that's where I'm stuck at the moment. I see the (ax+b)(cx+d)=y, but I've got nothing after that... I tried expanding it out which gives
y=acx^2+adx+bcx+bd and that yields nothing that I can see.
 
  • #6
If you don't know how to take a derivative and you have not been given a formula to tell when a line is tangent to a parabola, you can't do the problem.
 
  • #7
Ok, thanks anyway then. I guess I'll look for that formula, as I took a look at derivatives, and I doubt I could learn that without having a foundation in Calculus let alone Trig or Euclidian Geometry.
 
  • #8
You don't need calculus to find out when a line is tangent to a parabola. You just need to know how many times they intersect.
 
  • #9
Oh, right! It is also much simpler to do it from that perspective.
 

FAQ: Finding Tangent Lines for Parabolas: A Mathematical Approach

What is the meaning of "Y=x*y=x yields y=x^2 tangents"?

The statement "Y=x*y=x yields y=x^2 tangents" is a mathematical equation that represents a relationship between two variables, y and x. It means that when y and x are multiplied together, the result will be a parabola with the equation y=x^2, and the tangent line at any point on the parabola will have a slope of x.

How is this equation relevant to science?

This equation is relevant to science because it can be used to model various natural phenomena and physical systems. It is commonly used in physics, engineering, and other scientific fields to describe the relationship between two variables and predict their behavior.

What does the graph of this equation look like?

The graph of this equation, y=x^2, is a parabola that opens upwards. The tangent lines at different points on the parabola will have different slopes, but they will all pass through the point (0,0) on the graph.

How is the slope of the tangent line related to the value of x?

The slope of the tangent line at any point on the parabola is equal to the value of x at that point. This means that as x increases, the slope of the tangent line also increases, and as x decreases, the slope of the tangent line decreases.

What are some real-life applications of this equation?

This equation has many real-life applications, such as predicting the trajectory of a projectile, modeling the motion of objects under the influence of gravity, and describing the shape of a satellite's orbit around a planet. It is also used in fields like economics to model supply and demand curves, and in computer graphics to create 3D models and animations.

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