Question 107258: Could you please take the time to help me solve this problem: Sketch the following exponential functions: f(x) = 2x, g(x) = 3x, and h(x) = ex, where e is the natural number. Identify all intercept points as well as the behaviors of the functions when x is approaching plus and infinity.Thank you so much for helping me.
Thank you for helping me.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You are given the three functions:
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and
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You are to make the graphs of  ,  , and  .
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Some points to notice. Since e is approximately 2.718281828 for thinking about its graph you
could say it is the graph of  . Look at how similar these three graphs will
be ... one is the graph of ... another is ... and the last
one is
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So if you understand the procedure for one of the graphs, you can use that same procedure
on the other two graphs. Let's look at the graph of . We can get an idea
of its graph by picking values for x and calculating y. Suppose we let x = +1. What is
the corresponding value of y? Go to the equation and substitute +1 for x to get:
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So you can plot the point (+1,+2) as a point on the graph.
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Next let x = +2 and get the corresponding value of y:
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The point (2,4) is on the graph
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Next let x = +3 and get the corresponding value of y:
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The point (3,8) is on the graph
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One more ...
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Next let x = +4 and get the corresponding value of y:
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The point (4,16) is on the graph
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If you plot those points, you should have a pretty good idea of what is happening when
x is positive and is getting bigger.
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What about when x = 0? Any number raised to the exponent zero results in 1 as the answer.
Therefore, for all three of the graphs you are to make, when x is set equal to zero the
corresponding value of y is +1. Therefore (0, +1) is a point on all three of your graphs.
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Finally, what happens when x is a negative value? Remember that a negative exponent can
be converted to a positive exponent by putting the quantity in the denominator of a fraction
whose numerator is 1. Let's see what happens to when x is a negative
number.
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Begin by setting x = -1. The equation becomes:
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So the point (-1, 1/2) is on the graph.
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Next set x = -2. The equation becomes:
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So the point (-2, 1/4) is on the graph.
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Then set x = -3. The equation becomes:
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So the point (-3, 1/8) is on the graph.
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Next set x = -4. The equation becomes:
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So the point (-4, 1/16) is on the graph.
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By this time you can tell that as the value of x moves to the left in the negative
domain, the value of y gets smaller and smaller.
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You can apply this same methodology to the other two functions. [Note that a scientific
calculator can be used to find values of  but for your graphs you can just raise
2.72 to powers of x that you choose and that will be close enough to see what is going on
with the graph.]
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When you get done with the three graphs they should look like this:
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Where the "red" graph is the graph of and the "green" graph is the graph
of and the "purple" graph is the graph of .
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Notice that the "purple" graph is a path between the other two graphs.
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Hope this is enough info that you can work your way through the entire problem.
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