Question

compare the function $g(x) = 2|x + 1| - 4$ with $f(x) = |x|$. describe the graph of $g$ as a transformation of the graph of $f$. then graph the function.

describe the graph of $g$ as a transformation of the graph of $f$.

the graph of $f$ is translated \\(\square\\) unit(s) \\(\boldsymbol{\blacktriangledown}\\) and \\(\square\\) unit(s) \\(\boldsymbol{\blacktriangledown}\\) the graph \\(\boldsymbol{\blacktriangledown}\\) reflected across the x - axis and is \\(\boldsymbol{\blacktriangledown}\\) by a factor of \\(\square\\)
( type integers or decimals. )

Explanation:

Step1: Analyze horizontal translation

For the function \( f(x) = |x| \) and \( g(x) = 2|x + 1| - 4 \), the horizontal translation is determined by the change in the argument of the absolute value. The form \( |x - h| \) represents a translation \( h \) units horizontally. Here, we have \( |x + 1|=|x - (- 1)| \), so the graph of \( f(x) \) is translated 1 unit to the left.

Step2: Analyze vertical translation

The vertical translation is determined by the constant term at the end. The form \( a|x - h|+k \) has a vertical translation of \( k \) units. Here, \( k=-4 \), so the graph is translated 4 units down.

Step3: Analyze reflection and vertical stretch

The coefficient of the absolute value is 2, which is positive, so there is no reflection across the \( x \)-axis (reflection would be if the coefficient was negative). The coefficient 2 means the graph is vertically stretched by a factor of 2 (since \( |2|>1 \), it's a stretch, not a compression).

Answer:

The graph of \( f \) is translated 1 unit left and 4 units down. The graph is not reflected across the \( x \)-axis and is vertically stretched by a factor of 2. So filling in the blanks: 1, left, 4, down, is not, vertically stretched, 2.