, which of these statements describes the transformations to get the graph of for ?
Given the graphTranslate to the left by 1 and up by , then scale vertically by 2.
Translate to the left by and up by , then scale vertically by 2.
Translate to the left by 1 and up by , then scale vertically by 2.
Translate to the left by and up by , then scale vertically by 2.
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Let us work through the options
1) Translate the graph to the left by 1, translate the graph up by ln 4 , scale the graph vertically by 2.
The first transformation gives us y = ln ( x + 1 ) .
The second transformation gives us y = ln ( x + 1 ) + ln 4 .
The third transformation gives us y = 2 [ ln ( x + 1 ) + ln 4 ] = ln 1 6 ( x 2 + 2 x + 1 ) .
2) Translate the graph to the left by 2 1 , translate the graph up by ln 2 , scale the graph vertically by 2.
The first transformation gives us y = ln ( x + 2 1 ) .
The second transformation gives us y = ln ( x + 2 1 ) + ln 2 .
The third transformation gives us y = 2 [ ln ( x + 2 1 ) + ln 2 ] = ln ( 4 x 2 + 4 x + 1 ) .
3) Translate the graph to the left by 1, translate the graph up by ln 2 , scale the graph vertically by 2.
The first transformation gives us y = ln ( x + 1 ) .
The second transformation gives us y = ln ( x + 1 ) + ln 2 .
The third transformation gives us y = 2 [ ln ( x + 1 ) + ln 2 ] = ln ( 4 x 2 + 8 x + 4 ) .
4) Translate the graph to the left by 2 1 , translate the graph up by ln 4 , scale the graph vertically by 2.
The first transformation gives us y = ln ( x + 2 1 ) .
The second transformation gives us y = ln ( x + 2 1 ) + ln 4 .
The third transformation gives us y = 2 [ ln ( x + 2 1 ) + ln 4 ] = ln ( 1 6 x 2 + 1 6 x + 4 ) .
Hence, the answer is 2) Translate the graph to the left by 2 1 , translate the graph up by ln 2 , scale the graph vertically by 2.
Note: For the full graph of y = ln ( 4 x 2 + 4 x + 1 ) , we will also need to reflect it with respect to the line x = − 2 1 . The reason for this is that
y = ln ( 4 x 2 + 4 x + 1 ) = 2 [ ln 2 + ln ∣ x + 2 1 ∣ ] .