Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
Let's analyze the transformation of the parent function [tex]\( f(x) = |x| \)[/tex] into the given function [tex]\( g(x) = 12|x-7| \)[/tex] step-by-step.
### Step 1: Determine the Direction of the Opening
The direction of the opening of an absolute value graph is determined by the coefficient [tex]\( a \)[/tex]. If [tex]\( a \)[/tex] is positive, the graph opens upward. If [tex]\( a \)[/tex] is negative, the graph opens downward. In the function [tex]\( g(x) = 12|x-7| \)[/tex], the coefficient [tex]\( a = 12 \)[/tex], which is positive. Therefore, the graph opens upward.
### Step 2: Determine the Vertical Stretch
The vertical stretch of an absolute value graph is also determined by the coefficient [tex]\( a \)[/tex]. The vertical stretch factor is the absolute value of [tex]\( a \)[/tex]. In [tex]\( g(x) = 12|x-7| \)[/tex], the vertical stretch factor is [tex]\( |12| = 12 \)[/tex].
### Step 3: Determine the Horizontal Translation
The horizontal translation of an absolute value graph is determined by the value of [tex]\( h \)[/tex] inside the absolute value. For a function in the form [tex]\( g(x) = a|x-h| + k \)[/tex], the graph is translated [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive, and [tex]\( h \)[/tex] units to the left if [tex]\( h \)[/tex] is negative. In [tex]\( g(x) = 12|x-7| \)[/tex], the value of [tex]\( h = 7 \)[/tex], which means the graph is translated [tex]\( 7 \)[/tex] units to the right.
### Step 4: Determine the Vertical Translation
The vertical translation is determined by the value of [tex]\( k \)[/tex] in the function [tex]\( g(x) = a|x-h| + k \)[/tex]. If [tex]\( k \)[/tex] is positive, the graph is translated [tex]\( k \)[/tex] units upward, and if [tex]\( k \)[/tex] is negative, the graph is translated [tex]\( k \)[/tex] units downward. In [tex]\( g(x) = 12|x-7| \)[/tex], there is no [tex]\( k \)[/tex] term explicitly mentioned, so [tex]\( k = 0 \)[/tex], which means there is no vertical translation.
### Summary of the Transformations
- Direction of the Opening: Upward
- Vertical Stretch: By a factor of 12
- Horizontal Translation: [tex]\( 7 \)[/tex] units to the right
- Vertical Translation: None ( [tex]\( k = 0 \)[/tex])
Given these observations, option A correctly matches the analysis:
>A) The absolute value graph opens upward, has a vertical stretch by a factor of 12, and a horizontal translation 7 units right. There is no vertical translation.
Hence, the correct description of the transformation is indeed option A.
### Step 1: Determine the Direction of the Opening
The direction of the opening of an absolute value graph is determined by the coefficient [tex]\( a \)[/tex]. If [tex]\( a \)[/tex] is positive, the graph opens upward. If [tex]\( a \)[/tex] is negative, the graph opens downward. In the function [tex]\( g(x) = 12|x-7| \)[/tex], the coefficient [tex]\( a = 12 \)[/tex], which is positive. Therefore, the graph opens upward.
### Step 2: Determine the Vertical Stretch
The vertical stretch of an absolute value graph is also determined by the coefficient [tex]\( a \)[/tex]. The vertical stretch factor is the absolute value of [tex]\( a \)[/tex]. In [tex]\( g(x) = 12|x-7| \)[/tex], the vertical stretch factor is [tex]\( |12| = 12 \)[/tex].
### Step 3: Determine the Horizontal Translation
The horizontal translation of an absolute value graph is determined by the value of [tex]\( h \)[/tex] inside the absolute value. For a function in the form [tex]\( g(x) = a|x-h| + k \)[/tex], the graph is translated [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive, and [tex]\( h \)[/tex] units to the left if [tex]\( h \)[/tex] is negative. In [tex]\( g(x) = 12|x-7| \)[/tex], the value of [tex]\( h = 7 \)[/tex], which means the graph is translated [tex]\( 7 \)[/tex] units to the right.
### Step 4: Determine the Vertical Translation
The vertical translation is determined by the value of [tex]\( k \)[/tex] in the function [tex]\( g(x) = a|x-h| + k \)[/tex]. If [tex]\( k \)[/tex] is positive, the graph is translated [tex]\( k \)[/tex] units upward, and if [tex]\( k \)[/tex] is negative, the graph is translated [tex]\( k \)[/tex] units downward. In [tex]\( g(x) = 12|x-7| \)[/tex], there is no [tex]\( k \)[/tex] term explicitly mentioned, so [tex]\( k = 0 \)[/tex], which means there is no vertical translation.
### Summary of the Transformations
- Direction of the Opening: Upward
- Vertical Stretch: By a factor of 12
- Horizontal Translation: [tex]\( 7 \)[/tex] units to the right
- Vertical Translation: None ( [tex]\( k = 0 \)[/tex])
Given these observations, option A correctly matches the analysis:
>A) The absolute value graph opens upward, has a vertical stretch by a factor of 12, and a horizontal translation 7 units right. There is no vertical translation.
Hence, the correct description of the transformation is indeed option A.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
