Question

the graphs to the right show a vertical and a horizontal translation of ( f(x)=2^{x} ).
a. compare the graph of ( g(x)=2^{x + 3}+4 ) to the graph of ( f(x)=2^{x} ).
b. in general, when the graph of an exponential function is translated both vertically and horizontally, what is the effect on the asymptote?
c. in general, when the graph of an exponential function is translated both vertically and horizontally, what is the effect on the domain and range?
a. the graph of ( g ) is a translation unit(s) and unit(s) from the graph of ( f ).
(type whole numbers.)

Explanation:

Step1: Recall the rules of function translation

For a function \(y = f(x - h)+k\), \(h\) represents horizontal translation (if \(h>0\) shift right, if \(h < 0\) shift left) and \(k\) represents vertical translation (if \(k>0\) shift up, if \(k < 0\) shift down).
For \(g(x)=2^{x + 3}+4\), we can rewrite it as \(g(x)=2^{(x-(- 3))}+4\).

Step2: Determine the horizontal translation

Comparing \(g(x)=2^{(x-(-3))}+4\) with \(y = f(x - h)+k\) (where \(f(x)=2^{x}\)), for the horizontal part, \(h=-3\). Since \(h=-3<0\), the graph of \(f(x)=2^{x}\) is shifted \(|h| = 3\) units to the left.

Step3: Determine the vertical translation

For the vertical part, \(k = 4>0\). So the graph of \(f(x)=2^{x}\) is shifted \(k = 4\) units up.

Answer:

The graph of \(g\) is a translation \(3\) units to the left and \(4\) units up from the graph of \(f\).