Let f(x)=−9x+1. Match the function with the description. The graph of g is a reflection in the y-axis of the graph of f.
The graph of g is a reflection in the x-axis of the graph of f.
The graph of g is a horizontal translation 16 units right of the graph of f.
The graph of g is a vertical translation 16 units down of the graph of f.

I guess that we want to find the function g(x) for the 4 cases.

first, f(x) = -9*x + 1.

a) The graph of g is a reflection in the y-axis of the graph of f.

First remember: if we have the point (x,y) and we reflect it over the y-axis, we get (-x,y)

then g(x) = f(-x) = -9*-x + 1 = 9*x + 1.

b) The graph of g is a reflection in the x-axis of the graph of f.

if we have a point (x, y) and we reflect it over the x-axis, the point transforms into (x, -y)

then we have: g(x) = -f(x) = 9*x - 1

c) The graph of g is a horizontal translation 16 units right of the graph of f.

When we want to have a translation in the x-axis, we must change x by x - A.

If A is positive, this transformation moves the graph by A units to the right, in this case, A = 16.

g(x) = f(x - 16) = -9*(x - 16) + 1

d) The graph of g is a vertical translation 16 units down of the graph of f.

For vertical translations, if we want to move the graph by A units down (A positive) we should do y = f(x) - A

In this case, A = 16.

then: g(x) = f(x) - 16 = -9*x + 1 - 16 = -9*x - 15.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4