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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, left parenthesis, minus, x, right parenthesis, equals
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, left parenthesis, minus, x, right parenthesis, equals
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, left parenthesis, x, right parenthesis, 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, equals, E, left parenthesis, x, right parenthesis is shown below.
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, left parenthesis, minus, x, right parenthesis, equals, E, left parenthesis, x, right parenthesis for all x.

Interpreting symmetrical features

What does “E, left parenthesis, minus, x, right parenthesis, equals, E, left parenthesis, x, right parenthesis for all x” mean?
Because the statement is true for all x's, we can say that E, left parenthesis, minus, x, right parenthesis, equals, E, left parenthesis, x, right parenthesis is true when x, equals, 2, x, equals, 4, x, equals, 10, etc. Let's start by thinking about what the statement means for a specific x value, in this case when x, equals, 2.
When x, equals, 2, this statement becomes E, left parenthesis, minus, 2, right parenthesis, equals, E, left parenthesis, 2, right parenthesis.
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 start color #aa87ff, E, end color #aa87ff, left parenthesis, start color #1fab54, minus, 2, end color #1fab54, right parenthesis, equals, start color #aa87ff, E, end color #aa87ff, left parenthesis, start color #e07d10, 2, end color #e07d10, right parenthesis means that a start color #1fab54, start text, s, p, r, i, n, g, space, c, o, m, p, r, e, s, s, e, d, space, b, y, space, end text, 2, start text, space, m, e, t, e, r, s, end text, end color #1fab54 contains the same amount ofstart color #aa87ff, start text, space, e, n, e, r, g, y, end text, end color #aa87ff as start color #e07d10, start text, t, h, e, space, s, a, m, e, space, s, p, r, i, n, g, space, s, t, r, e, t, c, h, e, d, space, b, y, space, end text, 2, start text, space, m, e, t, e, r, s, end text, end color #e07d10.
What does E, left parenthesis, minus, 4, right parenthesis, equals, E, left parenthesis, 4, right parenthesis mean in this context?
Choose 1 answer:
Choose 1 answer:

We are now ready to interpret the more general statement, E, left parenthesis, minus, x, right parenthesis, equals, E, left parenthesis, x, right parenthesis, which is our ultimate goal.
Using the above examples for guidance, we see that E, left parenthesis, minus, x, right parenthesis, equals, E, left parenthesis, x, right parenthesis 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:
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, equals, T, left parenthesis, w, right parenthesis is shown below, where T, left parenthesis, w, right parenthesis indicates the change in the temperature of Pranav's house.

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, left parenthesis, minus, w, right parenthesis, equals, minus, T, left parenthesis, w, right parenthesis 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, left parenthesis, minus, w, right parenthesis, equals, minus, T, left parenthesis, w, right parenthesis” 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 start color #63d9ea, T, end color #63d9ea, left parenthesis, start color #ed5fa6, minus, 1, end color #ed5fa6, right parenthesis, equals, minus, start color #63d9ea, T, end color #63d9ea, left parenthesis, start color #7854ab, 1, end color #7854ab, right parenthesis means that the start color #63d9ea, start text, t, e, m, p, e, r, a, t, u, r, e, space, c, h, a, n, g, e, end text, end color #63d9ea that results from burning start color #ed5fa6, 1, start text, space, l, e, s, s, space, k, i, l, o, g, r, a, m, space, o, f, space, w, o, o, d, end text, end color #ed5fa6 is opposite that of what results from burning start color #7854ab, 1, start text, space, m, o, r, e, space, k, i, l, o, g, r, a, m, space, o, f, space, w, o, o, d, end text, end color #7854ab.
We are now ready to generalize and interpret the symmetry statement for a general w.
What does T, left parenthesis, minus, w, right parenthesis, equals, minus, T, left parenthesis, w, right parenthesis mean in this context?
Choose 1 answer:
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:
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, left parenthesis, x, right parenthesis, in miles per hour, is a function of the knob position, x. Note that x, is greater than, 0 means the knob is turned x units clockwise and x, is less than, 0 means the knob is turned x units counter-clockwise.
The graph of y, equals, V, left parenthesis, x, right parenthesis is shown below.
This function is even. Which of the following statements best interprets the symmetry of the graph of function V?
Choose 1 answer:
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?
    (9 votes)
    Default Khan Academy avatar avatar for user
    • 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.
      (40 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.
    (3 votes)
    Default Khan Academy avatar avatar for user
    • 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.
      (8 votes)
  • purple pi teal style avatar for user KKS
    Can f(x)=-f(x), except when f(x)=0
    (2 votes)
    Default Khan Academy avatar avatar for user
    • 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.
      (1 vote)
  • leaf blue style avatar for user Brian Bale
    So what would F(x)=-F(x) be?
    Even or neither.
    (1 vote)
    Default Khan Academy avatar avatar for user
  • 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.
    (1 vote)
    Default Khan Academy avatar avatar for user