Question

What effect does changing the function g(x)=logx to f(x)=log(x+3)+2 have on the graph of g(x)?

The graph shifts 3 units to the left and 2 units up.

The graph shifts 3 units to the right and 2 units down.

The graph shifts 2 units to the right and 3 units down.

The graph shifts 2 units to the left and 3 units up.

Answer 1

`\bf \begin{array}{llll} \bullet \textit{ vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\\\ \qquad if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{{{ B}}} \end{array}`

with that template in mind, let's see


`\bf \begin{array}{lllcclll} f(x)=&1log(&1x&+3)&+2\\ &\uparrow &\uparrow &\uparrow &\uparrow \\ &A&B&C&D \end{array} \\\\\\ \textit{horizontal shift by }\cfrac{C}{B}\implies \cfrac{3}{1}\implies +3\textit{ units to the left} \\\\\\ \textit{vertical shift by }D=+2\textit{ units upwards}`