first of all, SOLVE FOR EVERYTHING ALGEBRAICALLY. do NOTHING visually. only use the graph as a reality check. (ie, if you find one x-intercept, but there are clearly three, you know you did something wrong). Do solve for each one algebraically, do the following.
1. x-int:
This is asking for the x-intercept AKA the points where the graph touches the x-axis. At points where the graph touches the x-axis, we know that x=0 (by definition). Consequently, to solve for the x-intercepts, you simply plug x=0 into your equation and solve for y. Your x-intercepts will be coordinates of the form (0,y1) (0,y2) (0,y3)... where y1 is the first x-intercept, y2 is the second x-intercept y3... etc etc
2. y-int:
Just as before, this is asking for the y-intercept AKA the points where the graph touches the y-axis. Again, to find this, set y=0 and solve for x in your graph equation. Your answers will be coordinate points again.
3. vertical asymp:
A vertical asymptote is a point at which the graph shoots up or down towards infinity. In general, to find vertical asymptotes, you are going to find out what value of x "fucks your equation up." by "fuck the equation up" we generally mean something like dividing by zero. For instance, if your equation is y = 1/x. We clearly see that x=0 would result in a divide by zero, thus there is a vertical asymptote at x=0. Or, if our graph was y = 1/(1-x^2), we see that both x=1 and x=-1 result in a divide by zero. Therefore, we have vertical asymptotes x=1 and x=-1. Its important to note that "x = NUMBER" is the equation for a line. Your answer MUST be in the form "x = crap" or else your teacher will be pissed at you. (some students just say "we see asymptotes at 1 and 7" which is incorrect. you say "we have asymptotes at x=1 and x=7"
4. horiz. asymp:
For these, you just see what happens when you let x go towards infinity AND negative infinity. So, if your equation was y=1/x. We see that plugging in infinity yields y=0 and plugging in negative infinity yields y=0 as well. Therefore, we would have a horizontal asymptote at y=0. (its just a coincidence in my example that you got the same thing by plugging in both positive and negative infinity. always check for BOTH)
5. relative extrema:
relative extrema refer to the minimum and maximum points (peaks and valleys) along the curve. relative means "all points that are peaks or valleys." if the problem asked for the "absolute extrema" it would want the coordinates of the highest peak and the lowest valley. Since it just asks for relative extrema, this is pretty easy.
So, the derivative of the function tells you whether the function is increasing or decreasing at any point. So if y = x^2, we'd know y' = 2x. Therefore, at x=1, the function is increasing at a rate of 2. Likewise, at -2, the function is decreasing at a rate of 4. Therefore, since we are looking for peaks and valleys, these are points where the function ISNT increasing or decreasing, its just flat. In other words, these are the points where the derivative is ZERO.
So, to find your relative min/max, you simply take the derivative and set it equal to zero. This will give you the x-coordinates of the min/max (you'll need to find the corresponding y-coordinates by plugging the x-coordinates into the original function).
You're not done yet though!! For each point, you need to know whether its a minimum or a maximum. To do this, you use the second derivative. The second derivative tells you whether or not the function is concaved up or down (concave up = valley, concave down = peak). So, simply take an x-coordinate that's a min/max point, plug it into the SECOND derivative, and look at whether your answer is positive or negative. If the answer is positive, your function is concave up, and you have a minimum at your x-coordinate. if the answer is negative, your function is concaved down, and you have a maximum at your x-coordinate.
6. points of inflection:
If you are skiing down a mountain, the inflection point would be the point at which you'd say "OH SHIT" because its the point at which you are going the fastest. Using the peaks and valleys idea, if you start at the top of a peak, and start rolling down the side, you'll keep gaining speed as long as the hill is getting steeper and steeper. You'll start slowing down once you approach the valley and your hill is getting less steep. In other words, you are going the FASTEST right at the moment when the concavity changes (from concave up to concave down).
Therefore, to find the inflection points, take the SECOND derivative, and then set it equal to zero. Solving for x yields the x-coordinates of your inflection points. Again, plug these x-values back into the original function to find the coordinates of the inflection point
DONE EZ
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EPT![[image loading]](http://i30.tinypic.com/v8mql0.jpg)
