the graph shows that f(x)=3^x is translated horizontally and vertically to create the function g(x)=3^{x…

Answer:

2

Explanation:

Step1: Identify a key point on f(x).

The function is $f(x) = 3^x$. A key point on this graph is $(0, 1)$, since $f(0) = 3^0 = 1$. This point is visible on the blue curve.

Step2: Identify the corresponding point on g(x).

The function $g(x) = 3^{x-h} + k$ is a translation of $f(x)$. The horizontal asymptote of $f(x)$ is $y=0$. The graph shows the horizontal asymptote of $g(x)$ (red curve) is $y=2$. Thus, the vertical shift is $k=2$. The point $(0, 1)$ on $f(x)$ is shifted to $(0+h, 1+k)$ on $g(x)$. Substituting $k=2$, the corresponding point on $g(x)$ has a y-coordinate of $1+2=3$. From the graph of $g(x)$, the point with y-coordinate 3 is $(2, 3)$.

Step3: Determine the horizontal shift h.

The point $(0, 1)$ on $f(x)$ corresponds to the point $(2, 3)$ on $g(x)$. The horizontal shift is the difference in the x-coordinates: $h = 2 - 0 = 2$. This represents a shift of 2 units to the right. $$ (0, 1) \rightarrow (0+h, 1+k) = (2, 3) $$ $$ h = 2 $$ $$ k = 2 $$ The value of $h$ is 2.