Okay so this function is going to be the graph of x^2 shifted down by 4 units. Let's try and figure out what that is going to look like. First, let's find all the intercepts. The y-intercept occurs when x = 0. So if we plug that in we get: y = (0)^2 - 4 = -4. Thus, the y intercept occurs at -4. (Also notice that this point (0,-4) has the smallest y value. One way we can see that is because squares can never be negative, the least they can be is zero, so that means y has to be -4 or greater.) Now lets find the x intercepts, or where the graph intersects the x axis. Here y = 0. So we have 0 = x^2 - 4. So, x^2 = 4 --> x = +2 OR -2. The graph then looks like this:
A function is even if there is symmetry about the y axis, this means that f(x) = f(-x) for all x. A function is odd if there is symmetry about the origin, or equivalently f(x) = -f(-x). Let's see which one of these is true for this function. We know that the function, f(x) (which is the same thing as y), is: f(x) = x^2-4. Now f(-x) = (-x)^2 -4. But (-x)^2 = x^2 so f(-x) = x^2 - 4 = f(x) thus the function is even.
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