How to sketch y=2^(x+2) -1? - Detailed explanation please

1 Answer
Jan 20, 2018

Please see below.

Explanation:

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It is easier if we graph y=2^x as the base function first. This is an exponential function. To do this, we pick a few values for x and plug them into the function to find the corresponding y for each one:

x=1, y=2^1=2

x=2, y=2^2=4

x=3, y=2^3=8

x=oo, y=2^oo=oo

x=-1, y=2^-1=1/2^1=1/2

x=-2, y=2^-2=1/2^2=1/4

x=-3, y=2^-3=1/2^3=1/8

x=-oo, y=2^-oo=1/2^oo=1/oo=0

These results indicate that as x goes to oo, y goes to oo, and as x goes to -oo, y goes to 0, which means the y-axis is the horizontal asymptote. Plotting the points on a coordinate system gives us the graph:

graph{2^x [-10, 10, -5, 5]}

We know that if we add a constant to our x the graph shifts in the negative direction by that number of units.

We also know that if we subtract a constant from our y the graph will shift down by that number of units.

Therefore, if we add 2 to our x and subtract 1 from our y the graph will shift 2 units to the left and 1 unit down, and our function becomes:

y=2^(x+2)-1

The graph of this function is:

graph{2^(x+2)-1 [-10, 10, -5, 5]}