To graph an exponential function by hand, you need to find the intercept(s), plot a few additional points, and then connect the dots and draw the graph, using what you know of exponential behavior and the general shape of the curve.

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Remember that one side of an original exponential function will be a nearly horizontal line, while the other side will shoot upwards (or downwards) very quickly.

- Graph
*y*= 3^{x}

Since 3^{x} grows so quickly, I will not be able to find many reasonably-graphable points on the right-hand side of the graph. And 3^{x} will very quickly get very small on the left-hand side of the graph, so I probably won't find many useful plot-points there, either.

Instead, I will find a few plot-points in the middle, close to the origin, and then draw the graph from there.

Here is my T-chart:

While I have seven plot-points in my T-chart, only as many as five are reasonable to plot. So I plot them:

Plotted points:

I'd better not try to continue the line as a quadratic:

THIS GRAPH IS WRONG!

...or as a straight-ish or only vaguely curved line:

THIS GRAPH IS ALSO WRONG!

The exponential, remember, will get (and stay) very close to zero on the left-hand side, so I will draw the graph "skinnying along" the top of the *x*-axis on the left-hand side:

Drawing the left-hand side

And on the right-hand side, the exponential will get really big, so I'll draw it shooting up through the top of my graph:

Drawing the right-hand side

Then my graph for this exponential is:

Graph of *y* = 3^{2}

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