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Graph $f(x)=e^x$

I have no idea how to graph this. I looked on wolframalpha and it is just a curve. But how would I come up with this curve without the use of other resources (i.e. on a test).

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Why would plotting some sample points and connecting them not be good enough? – gt6989b Aug 21 '12 at 19:12
If you know how to graph $g(x)=\log(x)$, just reflect that graph around the line $y=x$, since $f$ and $g$ are inverses of each other. – gt6989b Aug 21 '12 at 19:12
It is a curve that comes up often in applications (exponential growth) and after a while it becomes not much less familiar than $y=x^2$. Informally, $e^x$ is steadily increasing, very close to $0$ when $x$ is large negative. It has value $1$ at $x=0$, and then grows very fast. – André Nicolas Aug 21 '12 at 19:14
Your statement that you "have no idea how to graph this" is really strange to me. I presume you know how to graph something, so you have some idea of how to graph $e^x$. So you would do better to explain why you think that the techniques you know will not work here, or what problems you encountered while trying to graph $e^x$. – MJD Aug 21 '12 at 19:19
If you're new to $e^x,$ then you should also know about the term: exponential growth. – user2468 Aug 21 '12 at 19:47
up vote 3 down vote accepted

You'd like to capture some of the important behavior of the graph, yes? For example, end-behavior (what happens as $x\to-\infty$ or $x\to\infty$), intercepts, where it's increasing/decreasing, any peaks or troughs.

In this case, you'll want to convey the idea that $f(x)$ is always positive, continuous, and increasing (should be evident from your sketch, and the fact that you'll not draw any jumps, peaks or troughs) You'll want to show that $f(0)=1$. You'll want to show that as $x\to-\infty$, $f(x)$ levels off and approaches $0$ asymptotically (again, should look that way from your sketch, but you may want to explicitly indicate the horizontal asymptote). You'll want to give the idea that as $x\to\infty$, $f(x)\to\infty$ (should look that way). You'll want to plot a few other points, too, like $f(1)=e\approx 2.8$ and $f(-1)=1/e\approx 0.35$. That should cover it pretty well.

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This helped a lot. Thanks! – Austin Broussard Aug 21 '12 at 19:29
I'm glad to help! – Cameron Buie Aug 21 '12 at 19:30
Since $e^x=1+x+\frac {x^2} 2 + \dots \frac {x^r}{r!} + \dots$, $e^x$ eventually grows at least as rapidly as (hence more rapidly than) $\frac {x^r} {r!}$ for any $r$. So it eventually "outstrips" any polynomial in $x$. The graph may look like "just a curve" - but it is one of the special curves of mathematics - after straight lines and conic sections. – Mark Bennet Aug 21 '12 at 20:38

You mention in your question that "I have no idea how to graph this" and "how would I come up with this curve without the use of other resources (i.e. on a test)". I know that you have already accepted one answer, but I thought that I would add a bit.

In my opinion, then best thing is to remember (memorize if you will) certain types of functions and their graphs. So you probably already know how to graph a linear function like $f(x) = 5x -4$. Before a test, you would ask your teacher which types of functions that you would be required to ne able to sketch by hand without any graphing calculator. Then you could go study these different types.

Now, if you don't have a graphing calculator, but you have a simple calculator, then you could also try to sketch the graph of a function by just "plotting points". So you draw a coordinate system with an $x$-axis and a $y$-axis and for various values of $x$ you calculate corresponding values of $y$ and then you plot those points. In the end you connect all the dots with lines/curves (so if the dots are all on a straight line, then you can just draw a straight line, but is things seem to curve one way or the other, then you try to "curve with the points").

No matter what, I would recommend to try and sit down with a piece of paper. Draw a coordinate system (maybe your a ruler). And try to sketch graphs of various functions. This will then familiarize you with how graphs of certain functions looks like.

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