Analyzing graphs! Question written below.

Question

Use complete sentences to analyze the graph of the function f(x) = log 1/4 x.

Part 1 [4 points]: Describe the domain, range, and general shape of this graph.

Part 2 [4 points]: Using complete sentences, explain how plotting specific points helps graph the function and note any critical points such as its intercepts.

Answer 1

We can rewrite f(x) = log[(1/4)x] as

 

log[(1/4)x] = y

 

The solution to a logarithm is the exponent.  The base number of a log is 10.

 

(1/4)x = 10^y

 

We know that when any number is raised to the zero power, the value becomes 1.   When a number is raised to the negative power, the value is the reciprocal of that number.

 

Lets chose x values such as -1, 0, and 1 to determine our domain and range.

 

 

(1/4)x = 10^y

(1/4)(-1) = 10^y

-1/4 = 10^y

 

y = log(-1/4) / log10

 

Our y value is not a real number when x is a negative value.

 

(1/4)(0) = 10^y

0 = 10^y

 

There is no exponent that will give you a result of 0 when x=0.

 

Your domain is (0, ∞).

 

If we keep performing test points to evaluate for y, we can obtain our range.

 

Since logarithms evaluate exponential function, the shape of the graph will concave down, but increasing because the solution of the graph is the exponent rather than a number raised to a power.