Question

graph $f(x) = -4|x + 1| - 3$.

Explanation:

Step1: Recall the vertex form of absolute value function

The general form of an absolute value function is \( f(x) = a|x - h| + k \), where \((h, k)\) is the vertex. For \( f(x)=-4|x + 1|-3 \), we can rewrite \( x + 1 \) as \( x-(-1) \), so \( h=-1 \) and \( k = - 3 \). So the vertex should be at \((-1,-3)\).

Step2: Analyze the transformation from \( y = |x| \)

• The coefficient \( a=-4 \): The negative sign reflects the graph over the \( x \)-axis, and the magnitude \( 4 \) vertically stretches the graph by a factor of \( 4 \).
• The \( h=-1 \) shifts the graph \( 1 \) unit to the left.
• The \( k = - 3 \) shifts the graph \( 3 \) units down.

Step3: Compare with the given graph

The given graph has its vertex at \((0,0)\), but the correct vertex for \( f(x)=-4|x + 1|-3 \) should be at \((-1,-3)\). Also, the slope and direction: since \( a=-4 \), the graph should open downward (because of the negative \( a \)) and be steeper (because \( |a| = 4>1 \)). The given graph opens upward, which is incorrect. To graph \( f(x)=-4|x + 1|-3 \):

1. Plot the vertex at \((-1,-3)\).
2. For \( x>-1 \), the function is \( f(x)=-4(x + 1)-3=-4x-4 - 3=-4x-7 \). When \( x = 0 \), \( f(0)=-7 \).
3. For \( x<-1 \), the function is \( f(x)=-4(-(x + 1))-3 = 4x + 4-3=4x + 1 \). When \( x=-2 \), \( f(-2)=-8 + 1=-7 \).
4. Draw two lines with slopes \( - 4 \) (for \( x>-1 \)) and \( 4 \) (for \( x<-1 \)) passing through the vertex \((-1,-3)\), opening downward.

Answer:

The given graph is incorrect. To graph \( f(x)=-4|x + 1|-3 \):

1. Vertex: Plot the point \((-1, - 3)\).
2. Right of vertex (\(x > - 1\)): Use the equation \(y=-4x - 7\). For example, when \(x = 0\), \(y=-7\), so plot \((0,-7)\) and draw a line with slope \(-4\) from \((-1,-3)\) through \((0,-7)\).
3. Left of vertex (\(x < - 1\)): Use the equation \(y = 4x+1\). For example, when \(x=-2\), \(y=-7\), so plot \((-2,-7)\) and draw a line with slope \(4\) from \((-1,-3)\) through \((-2,-7)\). The graph should open downward (because \(a=-4<0\)) and be vertically stretched by a factor of \(4\), shifted \(1\) unit left and \(3\) units down from the parent function \(y = |x|\).