Question

(Algebra II ) What is the equation of y = x^3 with the given transformations? vertical compression by a factor of 1/7, horizontal shift 8 units to the left, reflection across the x -axis.

Answer 1

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ \begin{array}{rllll} % left side templates f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}\mathbb{R}^{{{ B}}x+{{ C}}}+{{ D}} \end{array}$  $\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\ \bullet \textit{ vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\ \qquad if\ {{ D}}\textit{ is positive, upwards} \end{array}$

$\bf ----------------------------\\ f(x)=x^3\textit{compression by }\frac{1}{7}\quad or\quad A=7 \\ \quad \\ f(x)=7x^3\impliedby \textit{8 units to the left, or }C=+8 \\ \quad \\ f(x)=7(x+8)^3\impliedby \textit{now, reflection over x-axis, means} \\ \quad \\ f(x)=-7(x+8)^3\impliedby \textit{a negative leading term's coefficient}$