Subjects algebra

Exponential Translations 239E4B

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1. The problem asks which translation steps change the function $f(x) = 3^x$ to $g(x) = 3^{x+1} + 4$. 2. Recall that for exponential functions, adding inside the exponent shifts the graph horizontally, and adding outside shifts it vertically. 3. Specifically, $f(x + c)$ shifts the graph $c$ units to the left if $c > 0$, and to the right if $c < 0$. 4. Adding a constant $k$ outside the function, $f(x) + k$, shifts the graph vertically up by $k$ units if $k > 0$, and down if $k < 0$. 5. Here, $g(x) = 3^{x+1} + 4$ means the graph of $3^x$ is shifted left by 1 unit (because of $x+1$) and up by 4 units (because of $+4$). 6. Therefore, the correct translation is shifting $f(x)$ one unit to the left and four units up.