Question

1. given absolute value function below. write all transformations and graph the function. (3pts)

$f(x) = -|x - 4|$

Explanation:

Step1: Identify Parent Function

The parent function is \( y = |x| \), which has a V - shape with vertex at \((0,0)\), opening upwards.

Step2: Analyze Horizontal Shift

For the function \( f(x)=-|x - 4|\), the \( x-4 \) inside the absolute value indicates a horizontal shift. The rule for horizontal shift is: if we have \( y = |x - h|\), the graph shifts \( h \) units to the right (if \( h>0 \)) or \( h \) units to the left (if \( h < 0 \)). Here, \( h = 4 \), so the graph of \( y=|x| \) is shifted 4 units to the right.

Step3: Analyze Reflection

The negative sign in front of the absolute value, \( -|x - 4| \), indicates a reflection over the \( x \) - axis. The rule for reflection over the \( x \) - axis is: if we have \( y=-g(x) \), the graph of \( y = g(x) \) is reflected over the \( x \) - axis. So the graph of \( y = |x - 4| \) (after the horizontal shift) is reflected over the \( x \) - axis, which changes the direction of the opening from upwards to downwards.

Step4: Graphing the Function

1. Start with the parent function \( y = |x| \), vertex at \((0,0)\), opening up.
2. Shift the graph 4 units to the right: the vertex moves from \((0,0)\) to \((4,0)\), and the two arms of the V - shape are still opening upwards (before reflection).
3. Reflect the graph over the \( x \) - axis: the vertex remains at \((4,0)\), but now the V - shape opens downwards. To graph it, we can find some points. For \( x = 4 \), \( f(4)=-|4 - 4|=0 \). For \( x = 5 \), \( f(5)=-|5 - 4|=- 1 \). For \( x = 3 \), \( f(3)=-|3 - 4|=-1 \). Plot the vertex \((4,0)\), and the points \((3,-1)\) and \((5,-1)\), then draw the V - shape opening downwards.

Answer:

Transformations:

1. Horizontal shift: The graph of the parent function \( y = |x| \) is shifted 4 units to the right.
2. Reflection: The graph (after the horizontal shift) is reflected over the \( x \) - axis.

To graph \( f(x)=-|x - 4| \):

• The vertex is at \((4,0)\).
• The graph is a V - shaped graph opening downwards. Plot the vertex \((4,0)\), and two other points: when \( x = 3 \), \( y=-1 \); when \( x = 5 \), \( y=-1 \). Then draw the line segments connecting these points to form the V - shape.