Lesson Objectives:
- Determine whether a graph is symmetric with respect to the x-axis, y-axis, or the origin.- Determine whether a graph is even, odd, or neither even nor odd.
We can use information about symmetry to help us graph and analyze equations and functions.
Testing for Symmetry with Algebra
In order to test for symmetry with respect to the x-axis, we must replace with y with `-y`. If this results in an equivalent equation, then it is symmetric with respect to the x-axis.
In order to test for y-axis symmetry, we must replace x with `-x`. If this results in an equivalent equation, then the graph is symmetric with respect to the y-axis.
Then, for testing with respect to the origin, we must replace both the x and the y values with `-x` and `-y`. If this results in an equivalent equation, which is the same equation we started with, then it is symmetric with respect to the origin.
Graph the equation in a calculator and determine visually what kind of symmetry it has. Then test for symmetry with algebra.
So before we can graph this equation, we need to isolate the y. So we'll start by dividing both sides by 7. So we can cancel the 7 on the left, and we have `y = 5/7x^2-3/7`, which we can now graph in our calculator and get an open parabola which intersects the y-axis at (0, -3/7).
So you can visually see that this is symmetric with respect to the y-axis. Now we can test for y-axis symmetry by plugging in a `-x` into our original equation. So we have `7y = 5(-x)^2-3`. So whenever we raise a negative number to an even power, that gets rid of our negative sign. So this is equal to `7y = 5x^2-3`. Since this is identical to our original equation, we've confirmed that our equation is symmetric with respect to the y-axis.
Now let's check for x-axis symmetry. We need to plug in a `-y` into our original equation. So we have `7(-y) = 5x^2-3`. This is the same as `-7y = 5x^2-3`. As you can see this is not the same as our original equation. So this just shows algebraically that this does not have x-axis symmetry.
And then in order to check for symmetry to the origin, we have to plug in both a `-y` and a `-x` into our original equation. So this is the same as `-7y = 5x^2-3`. Since this is not the same as our original equation, we don't have symmetry with respect to the origin.
Graph the equation in a calculator and determine visually what kind of symmetry it has. Then test for symmetry with algebra.
So `x = y^2+4`. Let's start by isolating our y, so that we can graph it in our calculator. So if we subtract 4 from both sides, this cancels the 4 on our right, and then we have `x-4 = y^2`. Now if we take the square root of both sides, we'll have `sqrt(x-4) = y`. We can graph this in our calculator, and we'll get a parabola which opens to the right. Where this is our x-axis, and this is our y-axis. And you can visually see that it is symmetric with respect to the x-axis, since if we fold our graph over the x-axis, it would perfectly overlap.
We can determine symmetry using algebra, as well. Let's check for y-axis symmetry. We need to plug in `-x` in for x in our original equation. So that will have `-x = y^2+4`. Since this is not an equivalent statement to what we started with, the equation is not symmetric with respect to the y-axis.
Next, let's check for x-axis symmetry. By plugging in a `-y` for y, so that we get `x = (-y)^2+4`. Now `(-y)^2` is the same as `y^2`, so this is an equivalent statement to our original equation. So it is symmetric with respect to the x-axis.
And then to check for symmetry to the origin, we must plug in a `-x` and a `-y` into our original equation. So we have `-x = (-y)^2+4`, which is `-x = y^2+4`. This is not an equivalent statement to what we started with, so it is not symmetric with respect to the origin.
Even functions are symmetric with respect to the y-axis. So, `f(x)` is equivalent to `f(-x)`.
Odd functions are symmetric with respect to the origin, so `f(-x)` is equivalent to `-f(x)`.
And remember that many functions are neither even nor odd.
Determine whether the function is even, odd, or neither even nor odd.
`f(x) = -2x^3+3x`
So let's check if it's even. If it's even, then `f(x)` should equal `f(-x)`. `f(-x) = -2(-x)^3+3(-x)` which is equal to `+2x^3-3x`. Since this is not equivalent to our original statement, then we do not have an even function.
Now let's check if it's odd. If it's odd, then `f(-x)` should equal to `-f(x)`. `-f(x)` is equal to `-(-2x^3+3x)`, which is `2x^3-3x`. Since `2x^3-3x` is equivalent to `2x^3-3x`, then this statement is true, and we do have an odd function.
Determine whether the function is even, odd, or neither even nor odd.
`f(x) = 6x^2+3x^4-1`
Let's check if it's even. Does `f(x) = f(-x)`? So `f(-x)` is equal to `6(-x)^2+3(-x)^4-1`. Since the powers are all even, we have `6x^2+3x^4-1`.
This is equivalent to our original statement, so we have an even function.