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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.
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