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Symmetry of algebraic models

Learn how to interpret the symmetry of a graph in the context of an applied problem.

Introduction

In this article, we will learn how to interpret the symmetry of a graph in the context of an applied problem.
But first, let's refresh our memories regarding the symmetry of functions.

Symmetry of functions

Complete the missing sentences about even and odd functions.
When a function f has a graph that is symmetrical with respect to the y-axis, we say that f is
. Algebraically, this means that f(x)=
for every input value x.
When a function f has a graph that is symmetrical with respect to the origin, we say that f is
. Algebraically, this means that f(x)=
for every input value x.
It is
to have a function that is neither even nor odd.

Now, let's take a look at an example.

Example 1

The energy stored in a spring, E(x), in joules, is a function of the spring's displacement, x, in meters, from its relaxed state, with a positive x indicating a stretched spring and a negative x indicating a compressed spring. The graph of y=E(x) is shown below.
A convave up parabola on an x y coordinate plane that scales by two on the x axis and one on the y axis. It decreases from the interval negative infinity to zero and increases from zero to infinity. The vertex is at (zero, zero). It passes through points around (negative seven, two and one half) and (seven, two and one half).
What can we learn about the context from the symmetry of its graph?

The symmetry of function E

Let's apply what we know about symmetry to function E.
If you reflect the graph of function E over the y-axis, it lands on itself.
So, function E is an even function. Algebraically, this means that E(x)=E(x) for all x.

Interpreting symmetrical features

What does “E(x)=E(x) for all x” mean?
Because the statement is true for all x's, we can say that E(x)=E(x) is true when x=2, x=4, x=10, etc. Let's start by thinking about what the statement means for a specific x value, in this case when x=2.
When x=2, this statement becomes E(2)=E(2).
Focusing on what each variable represents can help with this interpretation. Remember that a positive input indicates a stretch of the spring and a negative input indicates a compression of the spring, and that an output represents the energy stored in the spring.
In this light, we see that E(2)=E(2) means that a spring compressed by 2 meters contains the same amount of energy as the same spring stretched by 2 meters.
What does E(4)=E(4) mean in this context?
Choose 1 answer:

We are now ready to interpret the more general statement, E(x)=E(x), which is our ultimate goal.
Using the above examples for guidance, we see that E(x)=E(x) means that a spring compressed by x meters contains the same amount of energy as a spring stretched by x meters.
In other words: A spring compressed by a certain amount stores the same amount of energy as a spring that has been stretched by that same amount.

Reflection question

A statement that interprets the symmetry of an even function does which of the following?
Choose 1 answer:

Let's try another example.

Example 2

Pranav normally uses 20 kilograms of wood per day in his wood stove to keep his house at 25 degrees Celsius. He tries adjusting the amount of wood, w, he burns to see how the temperature changes. Specifically, a positive w indicates an addition of w kilograms of wood and a negative w indicates a reduction of w kilograms of wood. The graph of y=T(w) is shown below, where T(w) indicates the change in the temperature of Pranav's house.
A cube root function on an x y coordinate plane that scales by one on the x axis and one on the y axis. The x axis is labeled w. Its middle point is at (zero, zero). It passes through points around (negative nine, negative seven), (negative one, negative two), (one, two), and (nine, seven).

The symmetry of function T

The graph of function T is symmetric with respect to the origin.
So, function T is an odd function. Algebraically, this means that T(w)=T(w) for all w.

Interpreting symmetrical features

To interpret the symmetry in this situation, we want to translate the mathematical statement “for any w-value, T(w)=T(w)” in terms of the context.
Again, let's start by thinking about the meaning of this for a particular w value. Then, we can go back and generalize.
To help with this, remember that a positive input indicates an addition of wood and a negative input indicates a reduction of wood, and that the function outputs a temperature change.
So we see that T(1)=T(1) means that the temperature change that results from burning 1 less kilogram of wood is opposite that of what results from burning 1 more kilogram of wood.
We are now ready to generalize and interpret the symmetry statement for a general w.
What does T(w)=T(w) mean in this context?
Choose 1 answer:

In other words: Increasing and decreasing the wood burned by a certain amount have exactly opposite effects on the temperature of the house.

Reflection question

A statement that interprets the symmetry of an odd function does which of the following?
Choose 1 answer:

Drawing a conclusion

In general, to interpret the meaning of the symmetry in the graph of a function, it is helpful to do the following:
Step 1: Decide if the function is even or odd and determine what this means algebraically.
Step 2: Understand what each variable represents in terms of the context.
Step 3: Come up with a statement that uses the meaning of the variables and compares the output values for opposite input values.

Try it yourself

Trudy is learning to drive a new kind of vehicle. The speed of the vehicle is determined by the position of a rotating knob. The vehicle's speed, V(x), in miles per hour, is a function of the knob position, x. Note that x>0 means the knob is turned x units clockwise and x<0 means the knob is turned x units counter-clockwise.
The graph of y=V(x) is shown below.
An absolute value function on an x y coordinate plane that scales by one on the x axis and one on the y axis. The y axis is labeled V. The vertex is at (zero, zero). From the interval negative infinity to zero, the graph decreases. From the interval zero to infinity, the graph increases. It passes through the points (negative eight, four), (negative two, one), (two, one), and (eight, four).
This function is even. Which of the following statements best interprets the symmetry of the graph of function V?
Choose 1 answer:

Want to join the conversation?

  • marcimus pink style avatar for user H. H.
    I answered the last question right, but I still don't understand why option A is wrong. "Adjacent knob positions result in speeds that are different by 1/2 miles per hour.". That seems correct to me. Both lines increase at that rate for each adjacent knob position, i.e., 1,2,3, etc. So what is wrong with that option?
    (12 votes)
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    • leaf green style avatar for user Mark Johnson
      The phrasing of the question is a bit tricky. It doesn't ask which statement is true, it asks which statement "best interprets the symmetry".

      The statement you selected may be true, but it doesn't say anything about the symmetry of the graph.

      The statement that "Rotating the knob clockwise and counter-clockwise by the same amount results in equal vehicle speeds" is not only true, but also explains what it means for this graph to be symmetrical.
      (49 votes)
  • blobby green style avatar for user Jerry Meng
    proof: any function can be written as sum of an even function and an odd function.
    (5 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Nice problem!

      For any function f(x), with a domain of all real numbers, define the functions
      g(x)=(f(x)+f(-x))/2
      h(x)=(f(x)-f(-x))/2.

      We have
      g(x)+h(x)
      =(f(x)+f(-x)+f(x)-f(-x))/2
      =2f(x)/2
      =f(x).

      So f(x) is the sum of g(x) and h(x).

      The function g(x) is even because

      g(-x)=(f(-x)+f(x))/2
      =(f(x)+f(-x))/2
      =g(x).

      The function h(x) is odd because

      h(-x)=(f(-x)-f(x))/2
      =-(f(x)-f(-x))/2
      =-h(x).

      Since f(x) is any arbitrary function, we conclude that any function, with a domain of all real numbers, can be written as the sum of an even function and an odd function.
      (14 votes)
  • blobby green style avatar for user jason.yth100
    Is there function that is both even or odd?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • purple pi teal style avatar for user KKS
    Can f(x)=-f(x), except when f(x)=0
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      Your statement does not make sense.
      f(x) = -f(x) would mean that each input "x" creates two output values y and -y and those two values are equal. The only situtation where y and -y would be equal is when y=0. You have excluded that value.

      Another issue with the statement is that a function can only generate one output for each input. Your statement is trying to indicate that there are two outputs for one input generated by the function. That would never happen.
      (4 votes)
  • leaf blue style avatar for user Brian Bale
    So what would F(x)=-F(x) be?
    Even or neither.
    (2 votes)
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  • leaf green style avatar for user qrrqtx
    While I understand that the point of the last section of the article is to illustrate the behavior of even functions, one could "assume" that "adjacent knob positions" means adjacent integer values, e.g. there's some sort of detent at each integer value. In that case, the first statement, "Adjacent knob positions result in speeds that are different by 1/2 mile per hour", would also be correct.
    (2 votes)
    Default Khan Academy avatar avatar for user