We will use inspection to find how we should begin to rewrite the function g(x)= 0.003x+2.3424x+12.2 to obtain the form g(x)= ax-h+k.
g(x)=0.003x+2.3424/x+12.2
Since the denominator of the given simple rational function is x+12.2, we want to rewrite its numerator in the form a( x+12.2)+ b for some a and b. Let's compare the appropriate expressions of numerators to each other.
0.003x+2.3424= a(x+12.2)+ b
Since the coefficients in front of x must be equal, we know that a= 0.003.
a(x+12.2)+ b
⇓
0.03(x+12.2)+ b
Next, the numerators, x, and 0.003(x+12.2)+ b, should be equal. Let's find b!
Therefore, we should rewrite the numerator of the given function in the following way.
0.003x+2.3424= 0.003(x+12.2)+ b
⇓
0.003x+2.3424= 0.003(x+12.2)+ 2.3058
Finally, let's rewrite the function g in the form g(x)= ax-h+k.
We are asked to describe the graph of g as a transformation of the graph of f(x)= ax. First, in our case a=2.3058.
f(x)=2.3058/x
Now we describe the transformation of graph f to obtain the graph of a rational function g.
Transformations of y=f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Notice the following.
f(x)=2.3058/x, g(x)=2.3058/x+ 12.2+ 0.003
⇓
g(x)=f(x+ 12.2)+ 0.03
Therefore, we should translate the graph of f 12.2 units to the left and 0.003 units up to get the graph of g.
