Question

Find the function rule for g(x). The function f(x) = x2 . The graph of g(x) is f(x) translated to the right 8 units and down 7 units. What is the function rule for g(x)? A. g(x) = (x + 7)2 + 8 B. g(x) = (x - 8)2 - 7 C. g(x) = (x - 7)2 - 8 D. g(x) = (x + 8)2 + 7

Answer 1

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


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

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

now, with that template in mind, let's see hmm g(x) to the right 8 units, meaning C/B is -8, so, you can just make C = -8 and B =1, -8/1 = -8

and down by 7 units, ok, that simply means D = -7

$\bf \begin{array}{lllll} g(x)=&1(&1x&+0)^2&+0\\ &\uparrow &\uparrow &\uparrow &\uparrow \\ &A&B&C&D\\ &\downarrow &\downarrow &\downarrow &\downarrow \\ g(x)=&1(&1x&-8)^2&-7 \end{array}\implies g(x)=1(1x-8)^2-7 \\\\\\ g(x)=(x-8)^2-7$