Showing Two Functions Are Symmetric About A Line

In summary, the two functions y_1 and y_2 are symmetric about the line y = \frac{c}{d} if the distance between y_1 and y, and the distance between y_2 and y, are the same.
  • #1
Bashyboy
1,421
5
Hello everyone,

I have the functions [itex]y_1 = \frac{c}{b} + d e^{-bx}[/itex] and[itex]y_2 = \frac{c}{b} - d e^{-bx} [/itex], where [itex]c \in \mathbb{R}[/itex], and [itex]b,d \in \mathbb{R}^+[/itex].
What I would like to know is how to show that these two functions are symmetric about the line [itex]y = \frac{c}{d}[/itex].

What I thought was that if y_1 and y_2 are symmetric about the line [itex]y = \frac{c}{d}[/itex], then the distance between y_1 and y, and the distance between y_2 and y, will be the same. That is, [itex]d_1 = \sqrt{(y_1 - y)^2 + (x - x_0)^2}[/itex] and [itex]d_2 = \sqrt{(y_2 - y)^2 + (x - x_0)^2}[/itex], where [itex]d_1 = d_2[/itex].

Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?
 
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  • #2
Bashyboy said:
Hello everyone,

I have the functions [itex]y_1 = \frac{c}{b} + d e^{-bx}[/itex] and[itex]y_2 = \frac{c}{b} - d e^{-bx} [/itex], where [itex]c \in \mathbb{R}[/itex], and [itex]b,d \in \mathbb{R}[/itex].
What I would like to know is how to show that these two functions are symmetric about the line [itex]y = \frac{c}{d}[/itex].

What I thought was that if y_1 and y_2 are symmetric about the line [itex]y = \frac{c}{d}[/itex], then the distance between y_1 and y, and the distance between y_2 and y, will be the same. That is, [itex]d_1 = \sqrt{(y_1 - y)^2 + (x - x_0)^2}[/itex] and [itex]d_2 = \sqrt{(y_2 - y)^2 + (x - x_0)^2}[/itex], where [itex]d_1 = d_2[/itex].

Is this a correct way of determining symmetry? Is it true in general? Are there any other ways in which I could prove symmetry?

Some comments:

(1) Your distance formula should not involve x, because for each x separately you want to show that the distance between the points (x,c/b) and (x,y_1(x)) is the same as the distance between the points (x,c/b) and (x,y_2(x)). The x drops out of these distance formulas (although, of course, they still contain y_1(x) and y_2(x)). After that, what you say would be correct.

(2) There is a much easier way.
 
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  • #3
And what might this easier method be, Ray?
 
  • #4
Bashyboy said:
And what might this easier method be, Ray?

That is for you to think about; I am not allowed to give solutions, nor would I want to. I can make one suggestion, however: think about what you would get if you drew graphs of the two functions on the same plot.
 
  • #5
I have already drawn the plot of these functions, and that was how I made inference I made, that the distances must be the same. I am not sure what else could be concluded from the plots.
 
  • #6
Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?
 
  • #7
Bashyboy said:
Would it perhaps be that the sum of the functions y1 and y2 is identically zero for all x, where is a real number?

Well, how would you write it after correcting your erroneous expressions given before?
 

FAQ: Showing Two Functions Are Symmetric About A Line

What does it mean for two functions to be symmetric about a line?

When two functions are symmetric about a line, it means that if you were to fold the graph of one function along the line of symmetry, the resulting graph would be identical to the other function.

How can I determine if two functions are symmetric about a line?

To determine if two functions are symmetric about a line, you can follow these steps:
1. Find the line of symmetry by setting the two functions equal to each other and solving for x.
2. Substitute the x-value of the line of symmetry into both functions to get the corresponding y-values.
3. If the y-values are equal, then the functions are symmetric about the line. If they are not equal, then the functions are not symmetric about the line.

What is the significance of two functions being symmetric about a line?

When two functions are symmetric about a line, it shows that they have a special relationship and can be used to understand each other more deeply. It also allows for easier analysis and comparison of the two functions.

Can two functions be symmetric about more than one line?

Yes, two functions can be symmetric about more than one line. A function can be symmetric about any vertical, horizontal, or diagonal line on a graph.

What are some real-life examples of two functions being symmetric about a line?

One real-life example of two functions being symmetric about a line is the motion of a pendulum. The position of the pendulum can be described by two functions, one for the horizontal displacement and one for the vertical displacement. These two functions are symmetric about the vertical line that passes through the point of suspension. Another example is the relationship between Celsius and Fahrenheit temperature scales, where the two functions are symmetric about the line y=x.

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