What roots can I use to sketch this graph of:
y = x^4 - x^2
To sketch the graph of the given equation y = x^4 - x^2, we can start by finding the roots or x-intercepts. The x-intercepts occur when the y-value is zero.
To find the roots, we set the equation equal to zero and solve for x:
x^4 - x^2 = 0
Factoring out x^2, we get:
x^2(x^2 - 1) = 0
Now, we have two factors:
x^2 = 0 (Case 1)
x^2 - 1 = 0 (Case 2)
Solving Case 1, we find:
x^2 = 0
x = 0
Therefore, x = 0 is a root.
In Case 2, we solve for x as follows:
x^2 - 1 = 0
Adding 1 to both sides:
x^2 = 1
Taking the square root of both sides:
x = ±1
Therefore, x = -1 and x = 1 are the other roots.
So, the roots of the equation y = x^4 - x^2 are x = 0, x = -1, and x = 1.