01.02 Graphs of Functions
Mathematics

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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 yintercept 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^24. 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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