Answer :
To solve this problem, we need to examine each transformation of [tex]\( f(x) = 2x - 6 \)[/tex] and match it with the corresponding description.
Let's start by analyzing each transformation:
1. [tex]\( g(x) = 8x - 24 \)[/tex]
\begin{itemize}
\item Comparing this with the original function [tex]\( f(x) \)[/tex], we see that the coefficient of [tex]\( x \)[/tex] (which is 2 in [tex]\( f(x) \)[/tex]) has been multiplied by 4 to become 8.
\item This indicates that the function has been stretched horizontally by a factor of 4 away from the y-axis.
\item The constant term has also changed from -6 to -24, which doesn't affect the horizontal stretching information.
\item Therefore, [tex]\( g(x) = 8x - 24 \)[/tex] corresponds to "stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the y-axis."
\end{itemize}
2. [tex]\( g(x) = 2x - 10 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -10, indicating that the function is shifted vertically.
\item Since -10 is 4 units less than -6, we have a vertical shift downward by 4 units.
\item Therefore, [tex]\( g(x) = 2x - 10 \)[/tex] corresponds to "shifts [tex]\( f(x) \)[/tex] 4 units down."
\end{itemize}
3. [tex]\( g(x) = 2x - 14 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -14, meaning a vertical shift.
\item Since -14 is 8 units less than -6, the function is shifted downward by 8 units.
\item This doesn't match any given description for this problem.
\end{itemize}
4. [tex]\( g(x) = 8x - 6 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] is changed to 8, suggesting a horizontal transformation.
\item This indicates that the function is stretched by a factor of 4 assuming the original function [tex]\( 2x \)[/tex] is stretched.
\item However, the constant term remains -6 which directly doesn't match any provided description.
\item This doesn't match any given description for this problem.
\end{itemize}
5. [tex]\( g(x) = 2x - 2 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -2, indicating a vertical shift.
\item Since -2 is 4 units more than -6, we have a vertical shift upward by 4 units.
\item This doesn't match any given description for this problem.
\end{itemize}
6. [tex]\( g(x) = 8x - 4 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] is 8, indicating a horizontal stretching.
\item The constant term changes from -6 to -4 as well, implying another transformation.
\item With both changes together, it does not directly match any simple described transformation.
\item This doesn't match any given description for this problem.
\end{itemize}
To summarize the matches:
- [tex]\( g(x) = 8x - 24 \)[/tex]: "stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the y-axis"
- [tex]\( g(x) = 2x - 10 \)[/tex]: "shifts [tex]\( f(x) 4 \)[/tex] units down"
Let's start by analyzing each transformation:
1. [tex]\( g(x) = 8x - 24 \)[/tex]
\begin{itemize}
\item Comparing this with the original function [tex]\( f(x) \)[/tex], we see that the coefficient of [tex]\( x \)[/tex] (which is 2 in [tex]\( f(x) \)[/tex]) has been multiplied by 4 to become 8.
\item This indicates that the function has been stretched horizontally by a factor of 4 away from the y-axis.
\item The constant term has also changed from -6 to -24, which doesn't affect the horizontal stretching information.
\item Therefore, [tex]\( g(x) = 8x - 24 \)[/tex] corresponds to "stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the y-axis."
\end{itemize}
2. [tex]\( g(x) = 2x - 10 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -10, indicating that the function is shifted vertically.
\item Since -10 is 4 units less than -6, we have a vertical shift downward by 4 units.
\item Therefore, [tex]\( g(x) = 2x - 10 \)[/tex] corresponds to "shifts [tex]\( f(x) \)[/tex] 4 units down."
\end{itemize}
3. [tex]\( g(x) = 2x - 14 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -14, meaning a vertical shift.
\item Since -14 is 8 units less than -6, the function is shifted downward by 8 units.
\item This doesn't match any given description for this problem.
\end{itemize}
4. [tex]\( g(x) = 8x - 6 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] is changed to 8, suggesting a horizontal transformation.
\item This indicates that the function is stretched by a factor of 4 assuming the original function [tex]\( 2x \)[/tex] is stretched.
\item However, the constant term remains -6 which directly doesn't match any provided description.
\item This doesn't match any given description for this problem.
\end{itemize}
5. [tex]\( g(x) = 2x - 2 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] remains 2, so there is no horizontal compression or stretching.
\item The constant term changes from -6 to -2, indicating a vertical shift.
\item Since -2 is 4 units more than -6, we have a vertical shift upward by 4 units.
\item This doesn't match any given description for this problem.
\end{itemize}
6. [tex]\( g(x) = 8x - 4 \)[/tex]
\begin{itemize}
\item The coefficient of [tex]\( x \)[/tex] is 8, indicating a horizontal stretching.
\item The constant term changes from -6 to -4 as well, implying another transformation.
\item With both changes together, it does not directly match any simple described transformation.
\item This doesn't match any given description for this problem.
\end{itemize}
To summarize the matches:
- [tex]\( g(x) = 8x - 24 \)[/tex]: "stretches [tex]\( f(x) \)[/tex] by a factor of 4 away from the y-axis"
- [tex]\( g(x) = 2x - 10 \)[/tex]: "shifts [tex]\( f(x) 4 \)[/tex] units down"
