How do you graph #f(x)=5^x#?

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Answer 1 · 2015-08-10T21:45:18

Find the intercepts and asymptotes, plot a few points, and then sketch the graph.

#f(x)=5^x#

Step 1. Find the domain and range.

#f(x)# is defined for all real values of #x#, so the domain is the set of all real numbers.

#5^x>0#, so the range is #f(x)>0#

Step 2. Find the #y#-intercept.

Let #x=0#.

#y= 5^0 = 1#

The #y#-intercept is at (#0,1#).

Step 3. Find the #x#-intercept.

There is no #x#-intercept, because the range is #f(x)>0#.

Step 4. Find the horizontal asymptote.

As #x# decreases without bound, #5^x# approaches #0#.

The horizontal asymptote is at #y=0#.

Step 5. Calculate some extra points.

We have one point: #f(0) =1#.

#f(1) =5^1 = 5#

#f(-1) = 5^(-1) = 1/5#

Step 6. Plot the axes and your three points.

Graph1

Step 7. Complete the graph with a smooth curve through the three points.

Graph2

And you have your graph.