Question

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

Answer 1

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\\\ \begin{array}{rllll} % left side templates f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}(\mathbb{R})^{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}} sin\left({{ B }}x+{{ C}} \right)+{{ D}}\\\\ f(x)=&{{ A}} log\left({{ B }}x+{{ C}} \right)+{{ D}} \end{array}\qquad$


$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative} \\\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \end{array}$

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

with that template in mind, let's see

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