Question

compare the graph to the absolute value parent function ($y = |x|$). select all the transformations that apply. graph of a transformed absolute value function with vertex at (2, 0), passing through (0, -4) and other points, and multiple - choice options for transformations: a reflected over the x - axis, b horizontal shift left, c horizontal shift right, d vertical shift up, e vertical shift down, f graph is narrower, g graph is wider

Explanation:

Step1: Analyze Reflection

The parent function \( y = |x| \) opens upwards. The given graph also opens upwards, so no reflection over x - axis (a is not applicable).

Step2: Analyze Horizontal Shift

The vertex of \( y = |x| \) is at \( (0,0) \). The vertex of the given graph is at \( (2,0) \), so it has shifted 2 units to the right (c is applicable, b is not).

Step3: Analyze Vertical Shift

The vertex of the given graph is at \( y = 0 \), same as the parent function's vertex y - coordinate, so no vertical shift (d and e are not applicable).

Step4: Analyze Width (Stretch/Compression)

For \( y=|x| \), the slope of the right - hand side is 1. For the given graph, let's find the slope. Taking two points on the right - hand side, say \( (2,0) \) and \( (3, - 1) \), the slope \( m=\frac{-1 - 0}{3 - 2}=- 1 \) (the magnitude of the slope is 1, same as \( y = |x| \)). Wait, no, let's check another way. The general form of absolute value function is \( y=a|x - h|+k \). Here \( h = 2,k = 0 \), and let's take a point \( (0,-4) \) on the graph. Plugging into \( y=a|x - 2| \), we get \( - 4=a|0 - 2|\), \( - 4 = 2a\), \( a=-2 \). Wait, the slope of the right - hand side for \( y = |x| \) is 1, for \( y=-2|x - 2| \), the slope of the right - hand side (when \( x\geq2 \)) is \( - 2 \), so the magnitude of the slope is 2, which is greater than 1. So the graph is narrower (f is applicable, g is not). Wait, earlier mistake: when \( a>1 \) or \( a < - 1 \), the graph is narrower. Since \( |a| = 2>1 \), the graph is narrower. Also, the horizontal shift: vertex at \( (2,0) \), so horizontal shift right (c), and since \( |a|=2>1 \), the graph is narrower (f). Also, wait, the y - intercept of the parent function \( y = |x| \) is 0, the y - intercept of the given graph: when \( x = 0 \), \( y=-2|0 - 2|=-4 \), but the vertical shift: the vertex y - coordinate is 0, same as parent, but the graph is stretched vertically (narrower) and shifted horizontally right.

Wait, let's re - evaluate the slope. For \( y = |x| \), at \( x = 1 \), \( y = 1 \); at \( x = 2 \), \( y = 2 \). For the given graph, at \( x = 2 \), \( y = 0 \); at \( x = 3 \), \( y=-2 \); at \( x = 1 \), \( y=-2 \). So the rate of change (slope magnitude) is 2, which is greater than 1, so the graph is narrower (f). And the vertex is at \( x = 2 \), so horizontal shift right (c). Also, is there a reflection? The graph opens downward? Wait, the parent function \( y = |x| \) opens upward. The given graph: when \( x<2 \), the slope is positive (from \( (2,0) \) to \( (0,-4) \), slope is \( \frac{-4 - 0}{0 - 2}=2 \)), when \( x>2 \), slope is negative (from \( (2,0) \) to \( (4,-4) \), slope is \( \frac{-4 - 0}{4 - 2}=-2 \)). Wait, the graph opens downward? Wait, the vertex is at \( (2,0) \), and the graph goes down from the vertex, so it is a reflection over the x - axis? Wait, I made a mistake earlier. The parent function \( y = |x| \) opens upward (vertex is the minimum point). The given graph has the vertex as the maximum point, so it is reflected over the x - axis (a is applicable). Let's recalculate:

For the given graph, let's find the equation. Vertex at \( (2,0) \), and it opens downward. So the equation is \( y=-a|x - 2| \). Take the point \( (0,-4) \): \( - 4=-a|0 - 2|\), \( - 4=-2a \), \( a = 2 \). So the equation is \( y=-2|x - 2| \).

Now, re - analyzing:

- Reflection: Since \( a=-2 \) (negative), it is reflected over the x - axis (a is applicable).
- Horizontal shift: Vertex at \( x = 2 \), so shift right 2 units (c is applicable).
- Vertical shift: Vertex at \( y = 0 \), same as parent, so no vertical shift (d and e not applicable).
- Width: \( |a| = 2>1 \), so the graph is narrower (f is applicable, g not applicable).

Answer:

a. Reflected over the x - axis, c. Horizontal Shift Right, f. Graph is Narrower