9. the graph of the logarithmic function (f(x)) is shown. graph the function (k(x)=f(x + 6)+4) on the same…

Explanation:

Step1: Recall function - transformation rules

For a function (y = f(x)), (y=f(x + h)+k) is a horizontal shift by (h) units (left if (h>0), right if (h < 0)) and a vertical shift by (k) units (up if (k>0), down if (k < 0)). Here (h = 6) and (k = 4), so the graph of (y = f(x)) is shifted 6 units to the left and 4 units up to get (k(x)=f(x + 6)+4).

Step2: Analyze reference - point transformation

Let a reference point on (y = f(x)) be ((x_0,y_0)). The corresponding point on (k(x)) will be ((x_0-6,y_0 + 4)). For example, if a point on (f(x)) is ((1,0)) (assuming from the general shape of a logarithmic - type graph), the corresponding point on (k(x)) is ((1 - 6,0 + 4)=(-5,4)).

Step3: Determine asymptotes

For a logarithmic function (y = f(x)=\log(x)) (assuming the basic form of the given (f(x))), the vertical asymptote is (x = 0). For the function (k(x)=f(x + 6)+4), the vertical asymptote is found by setting (x+6=0), so (x=-6). The horizontal asymptote of a basic logarithmic function (y = f(x)) does not exist (it is a non - rational function with a vertical asymptote). Since we are only shifting the function vertically and horizontally, the non - existence of the horizontal asymptote remains the same for (k(x)).

a.

To transform each reference point on (f(x)) to create the graph of (k(x)), we subtract 6 from the (x) - coordinate and add 4 to the (y) - coordinate of each reference point on (f(x)).

b.

The vertical asymptote of (f(x)) is (x = 0). The vertical asymptote of (k(x)) is (x=-6). Neither (f(x)) nor (k(x)) has a horizontal asymptote.