The graphs below have the same shape. What is the equation of the red graph?

Accepted Solution

[tex]\bf ~\hspace{10em}\textit{function transformations} \\\\\\ \begin{array}{llll} f(x)= A( Bx+ C)^2+ D \\\\ f(x)= A\sqrt{ Bx+ C}+ D \\\\ f(x)= A(\mathbb{R})^{ Bx+ C}+ D \end{array}\qquad \qquad \begin{array}{llll} f(x)=\cfrac{1}{A(Bx+C)}+D \\\\\\ f(x)= A sin\left( B x+ C \right)+ D \end{array} \\\\---------------------------------[/tex][tex]\bf \bullet \textit{ stretches or shrinks horizontally by } A\cdot B\\\\ \bullet \textit{ flips it upside-down if } A\textit{ is negative}\\ ~~~~~~\textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if } B\textit{ is negative}\\ ~~~~~~\textit{reflection over the y-axis} \\\\ \bullet \textit{ horizontal shift by }\frac{ C}{ B}\\ ~~~~~~if\ \frac{ C}{ B}\textit{ is negative, to the right}[/tex][tex]\bf ~~~~~~if\ \frac{ C}{ B}\textit{ is positive, to the left}\\\\ \bullet \textit{ vertical shift by } D\\ ~~~~~~if\ D\textit{ is negative, downwards}\\\\ ~~~~~~if\ D\textit{ is positive, upwards}\\\\ \bullet \textit{ period of }\frac{2\pi }{ B}[/tex]
keeping that template in mind, the g(x) function is just a translated version of f(x), it is translated upwards by 2 units, namely D = 2.[tex]\bf f(x)=3-x^4\implies f(x)=\stackrel{A}{-1}(\stackrel{B}{1}x+\stackrel{C}{0})^4+\stackrel{D}{3}\implies \stackrel{\textit{shifted upwards by 2 units}}{f(x)=\stackrel{A}{-1}(\stackrel{B}{1}x+\stackrel{C}{0})^4+\stackrel{D}{3+2}} \\\\\\ f(x)=-1(1x+0)^4+5\implies f(x)=-x^4+5\implies f(x)=5-x^4=g(x)[/tex]