Exponential Functions: Graphing

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Exponential Functions: Graphing (page 3 of 5)

Sections: Introduction, Evaluation, Graphing, Compound interest, The natural exponential

Graph h(x) = (¹/₃ )^–x.

I will compute the same points as previously:

T-chart for h(x) = (1/3)^(-x)

Then the graph is:

graph of (1/3)^(-x) matches that of 3^x

So y = (¹/₃ )^–x does indeed model growth. For the record, however, the base for exponential functions is usually greater than 1, so growth is usually in the form "3^x" (that is, with a "positive" exponent) and decay is usually in the form "3^–x" (that is, with a "negative" exponent).

Every once in a while they'll give you a more-complicated exponential function to deal with:

Graph y = e^x^².

I will compute some plot-points, as usual:

T-chart for y = e^(x^2)

Note that, for graphing, the decimal approximations are more useful than the "exact" forms. For instance, it is hard to know where "e^2.25" should be plotted, but it's easy to find where "9.488" goes. Note also that I calculated more than just whole-number points. The exponential function grows way too fast for me to use a wide range of x-values (I mean, look how big y got when x was only 2). Instead, I had to pick some in-between points in order to have enough reasonable dots for my graph.

Now I plot the points, and draw my graph:

graph of y = e^(x^2)

The above function did involve an exponential, but was not in the "usual" exponential form (since the power was not linear, but quadratic). However, usually you'll get the more-standard form, with a greater-than-one base, perhaps multiplied by some constant, and a linear exponent. Note that the graphs all look roughly the same; they may have been moved up or down, flipped upside-down, shifted left or right, etc, etc, but they all pretty much have the same shape: