Other relevant information would include the definitions of "even" and "odd" functions. How are these terms defined?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
Have you graphed this equation? The graph would show that this parabola is not symmetric about the line x = 6.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.
Mark44 said:Other relevant information would include the definitions of "even" and "odd" functions. How are these terms defined?
Have you graphed this equation? The graph would show that this parabola is not symmetric about the line x = 6.
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).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.
Yes, and that will get the quadratic function in vertex form, which will be useful for answering the question about the symmetry.coco richie said:I'm thinking that I have to complete the square?
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.
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.
The equation for determining if a function is even or odd is:
Even function: f(x) = f(-x)
Odd function: -f(x) = f(-x)
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.
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.