Symmetry of f(x) = x^2 - 6x: Even or Odd?

In summary, we discussed how to determine if a function is symmetrical and whether it is even or odd. We looked at the definitions of even and odd functions, as well as how to identify them through graphing or function notation. We also mentioned using completing the square to find the vertex of a parabola and how it can be helpful in determining symmetry. Finally, we briefly touched on finding the axis of symmetry for a function.
  • #1
coco richie
8
0

Homework Statement



How can I tell if the following equation is symmetrical and whether it is even or odd?

Homework Equations



f(x) = x^2 - 6x

The Attempt at a Solution



I would guess that it is symmetric about x = 6 but that's not what the book says.
 
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  • #2
coco richie said:

Homework Statement



How can I tell if the following equation is symmetrical and whether it is even or odd?

Homework Equations



f(x) = x^2 - 6x
Other relevant information would include the definitions of "even" and "odd" functions. How are these terms defined?
coco richie said:

The Attempt at a Solution



I would guess that it is symmetric about x = 6 but that's not what the book says.
Have you graphed this equation? The graph would show that this parabola is not symmetric about the line x = 6.
 
  • #3
Might be cheating but if you know how to find the minimum on the graph, as (I tell you) there is only one, IF if is symmetrical it would have to be symmetrical about that, do you see that?
 
  • #4
Mark44 said:
Other relevant information would include the definitions of "even" and "odd" functions. How are these terms defined?

The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the origin.

Have you graphed this equation? The graph would show that this parabola is not symmetric about the line x = 6.

I'm thinking that I have to complete the square?
 
  • #5
coco richie said:
The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the origin.
I think Mark44 is asking for another definition, one that involves function notation (ie. f(x), or, to give you a hint, f(-x)). Look in your book and try again.EDIT: I was right (see below).
 
  • #6
There are definitions of these terms that don't involve the graphs.
A function is even if f(-x) = f(x) for all x in the domain of f.
A function is odd if f(-x) = -f(x) for all x in the domain of f.

Completing the square will give you information about the vertex of this parabola.
 
  • #7
coco richie said:
I'm thinking that I have to complete the square?
Yes, and that will get the quadratic function in vertex form, which will be useful for answering the question about the symmetry.
 
  • #8
Or another way. f(x) = 0 , for your f is very easy to solve - your f has been factorised already for you. So you have got a value of f for which there are two points, two values of x. Call them x1 and x2 to be more general. So if f is symmetrical it has to be symmetrical for these two points. I.e. it has to be symmetrical around what x-value (call it xm)? That would be true for any function f. The converse is not true for any function in general but it is true in general that an axis of symmetry can't be any other value of x than xm the one found as above*, so you will narrow it down vastly by finding that. Once you have you can probably show that x = xm is a line of symmetry for your whole function in your case (sufficiency as well as necessity).

To know what I'm talking about it may be better to draw a graph of you function. Quite an important problem because it is the start of more general and extensive principles.

*i.e. the condition is necessary but not sufficient
 

FAQ: Symmetry of f(x) = x^2 - 6x: Even or Odd?

What does it mean for a function to be even or odd?

When a function is even, it means that it has symmetry about the y-axis, meaning that if you were to fold the graph in half along the y-axis, the two halves would match up perfectly. On the other hand, when a function is odd, it has symmetry about the origin, meaning that if you were to rotate the graph 180 degrees around the origin, it would match up with its original position.

How can I determine if a function is even or odd?

To determine if a function is even or odd, you can use the following test:
Even function: Replace x with -x in the function. If the resulting function is the same as the original, then the function is even.
Odd function: Replace x with -x in the function. If the resulting function is the negative of the original, then the function is odd.

What is the equation for determining if a function is even or odd?

The equation for determining if a function is even or odd is:
Even function: f(x) = f(-x)
Odd function: -f(x) = f(-x)

Is the function f(x) = x^2 - 6x an even or odd function?

To determine if the function is even or odd, we can use the above test.
Even function: f(-x) = (-x)^2 - 6(-x) = x^2 + 6x
Since f(-x) is not equal to the original function, f(x) = x^2 - 6x is not an even function.
Odd function: -f(-x) = -(x^2 - 6x) = -x^2 + 6x
Since -f(-x) is equal to the original function, f(x) = x^2 - 6x is an odd function.

How does the symmetry of a function affect its graph?

The symmetry of a function affects its graph by creating certain patterns and relationships between points on the graph. For example, an even function will have a graph that is symmetrical about the y-axis, meaning that points on the left side of the y-axis will have corresponding points on the right side. Similarly, an odd function will have a graph that is symmetrical about the origin, with points on the top half of the graph having corresponding points on the bottom half. The symmetry can also help in determining the behavior of a function, such as the location of its minimum or maximum points.

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