SOLUTION: Describe the trasformations on the following graph: g(x)=-log(x)+2, give description of trasformation, equation for the vertical asymptote and the x-intercept in (x, y) form.

Question 711916: Describe the trasformations on the following graph: g(x)=-log(x)+2, give description of trasformation, equation for the vertical asymptote and the x-intercept in (x, y) form.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Starting with the graph of the "base" function of log(x):

The "-" in front of the log will cause a reflection of the graph in the x-axis.
The "+2" will cause a vertical translation, up 2.

The vertical asymptote, reflected in the x-axis (a vertical reflection) and then translated up 2, will still be the same! So the vertical asymptote of log(x), x = 0, will also be the vertical asymptote of g(x).

An x-intercept by definition is on the x-axis. All points on the x-axis have a y coordinate of 0. So to find an x-intercept make the y be 0 and solve for x:

Subtract 2:

Divide (or multiply) by -1:

Since the base of "log" is 10 this equation tells us that x is what you get if you raise 10 to the 2nd power, i.e. 100:

So the x-intercept is (100, 0).

Here's a look at the graphs of log(x) (in red), -log(x) (in green) and -log(x)+2 (in blue). Note the transformations. Also, Algebra.com's graphing software is not perfect. All three graphs look like they intersect the y-axis. They do not. The y-axis, x=0, is the vertical asymptote!

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