Graphing Exponential Functions: Examples (page 3 of 4) Sections: Introductory concepts, Step-by-step graphing instructions, Worked examples
This is the standard exponential, except that the "+ 4" pushes the graph up so it is four units higher than usual.
I need to remember that the "negative" exponent reverses the location (along the x-axis) in which the power on 5 is negative. When the x-values are negative (that is, when I'm on the left-hand side of the graph), the value of –x will be positive, so the graph will grow quickly on the left-hand side. On the other hand, when the x-values are positive (that is, on the right-hand side of the graph), the value of –x will be negative, so the graph will stay very close to the x-axis.
Any graph that looks like the above (big on the left and crawling along the x-axis on the right) displays exponential decay, rather than exponential growth. For a graph to display exponential decay, either the exponent is "negative" or else the base is between 0 and 1. You should expect to need to be able to identify the type of exponential equation from the graph. The first two worked examples displayed exponential growth; the last example above displays exponential decay; and the following displays exponential growth again. This is not the same as "2x + 3". In "2x + 3", the standard exponential is shifted up three units. In this case, the shift in "inside" the exponential. Instead of the "+ 3" shifting the "2x" up by three, the "+ 3" shifts the "2x" over sideways by three. The only question is: shifts sideways which way, left or right? The way I keep it straight is to consider one of the basic points on any exponential. When the power is zero, the exponential is 1. For "2(x + 3)", when is the power zero? When x + 3 = 0, so x = –3. That is, the basic plot point (0, 1) has been shifted to the point (–3, 1), so the graph has been shifted three points to the left:
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