Determine whether the graph of x=y^2-1 is symmetric with respect to the x-axis, the y-axis, the origin, or has no symmetry.
about x axis?
for any x is are there equal in magnitude + and - values of y?
y ^2 = x+1
y = +/- sqrt (x+1) so yes but x must be >-1
about y axis?
Nah, this is a parabola opening to the right with as vertex at (-1 , 0)
The problem is a quadratic function so that tells you right there that it has symmetry. Now you just have to find where.
Because the problem says x= and not y=
our problem with be rotated to the right
Its rotated to the right because y can be any value whether positive or negative but x will always be a positive number because y is squared.
So now our graph is quadratic and rotated right.(Like a sideways U shape)
Now it says -1 after y^2. This means the graphs shifting down one on the x axis.
With all this being said that means it is symmetrical about the x-axis
Yes, what Mia said :)
To determine the symmetry of a graph, we need to analyze the equation and identify any patterns or transformations. In this case, we have the equation x = y^2 - 1.
To test for symmetry with respect to the x-axis, we will replace y with -y in the equation and see if the resulting equation is equivalent to the original equation. If it is, then the graph is symmetric with respect to the x-axis.
Let's replace y with -y in the equation and simplify:
x = (-y)^2 - 1
x = y^2 - 1
As we can see, the equation remains the same. Therefore, the graph is symmetric with respect to the x-axis.
Next, we will test for symmetry with respect to the y-axis. To do this, we replace x with -x in the equation and check if the equation remains the same. If it does, then the graph is symmetric with respect to the y-axis.
Let's replace x with -x in the equation and simplify:
-x = y^2 - 1
Rearranging this equation, we get:
y^2 = x + 1
We can see that the equation has changed, which means the graph is not symmetric with respect to the y-axis.
Now, let's test for symmetry with respect to the origin. To do this, we replace both x and y in the equation with their negative values (-x and -y) and see if the equation remains unchanged. If it does, then the graph is symmetric with respect to the origin.
Replacing x with -x and y with -y in the equation, we have:
-x = (-y)^2 - 1
-x = y^2 - 1
This is the same equation as we started with, which means the graph is symmetric with respect to the origin.
In summary, the graph of x = y^2 - 1 is symmetric with respect to the x-axis and the origin, but not symmetric with respect to the y-axis.