Which absolute value equation creates the lower function in the graph? Be sure to consider the scales of the x- and y-axis when evaluating the question. (1 point) Responses y=2|x|−2 y equals 2 Start Absolute Value x End Absolute Value minus 2 y=12|x|−4 y equals Start Fraction 1 over 2 End Fraction Start Absolute Value x End Absolute Value minus 4 y=|x−4|−1 y equals Start Absolute Value x minus 4 End Absolute Value minus 1 y=|x|−4

Question

I can't describe the graph well but how would it look like for every answer?

Answer 1

Certainly! Let's go through each of the absolute value equations you've provided and describe how each graph would look, as well as give a brief explanation for why each graph is positioned at a different level vertically.

1. \( y = 2|x| - 2 \):

   • This function has a "V" shape that opens upwards, similar to the basic absolute value function.
   • The vertex is at (0, -2). The graph stretches vertically by a factor of 2 (because of the 2 in front of |x|), meaning it is steeper than the standard \( y = |x| \).
   • As x moves away from 0, the y-value increases steeply.

2. \( y = \frac{1}{2}|x| - 4 \):

   • This graph also has a "V" shape that opens upwards.
   • The vertex is at (0, -4). The slope of the lines is less steep than the standard absolute value function because of \( \frac{1}{2} \), which makes it wider.
   • This graph is lower than both of the previous ones since its lowest point is at -4.

3. \( y = |x - 4| - 1 \):

   • This graph has a "V" shape that opens upwards.
   • The vertex is at (4, -1). The graph shifts to the right by 4 units and down by 1 unit.
   • This graph is positioned above the graph of \( y = \frac{1}{2}|x| - 4 \) due to its vertex being at -1.

4. \( y = |x| - 4 \):

   • This graph has a "V" shape that opens upwards.
   • The vertex is at (0, -4). While it has the same lowest point as \( y = \frac{1}{2}|x| - 4 \), it is steeper than that graph.
   • Therefore, this graph, while starting at -4, will increase more rapidly than the \( \frac{1}{2}|x| - 4 \) graph.

Summary of their ranges:

• \( y = 2|x| - 2 \): Lowest point at (0, -2), f(x) increases from -2.
• \( y = \frac{1}{2}|x| - 4 \): Lowest point at (0, -4), but is wide, making it stay below -2 until much larger values of |x|.
• \( y = |x - 4| - 1 \): Lowest point at (4, -1), remains above -1 for all x.
• \( y = |x| - 4 \): Lowest point at (0, -4) but with a steeper slope.

In terms of identifying which equation creates the lowest graph, \( y = \frac{1}{2}|x| - 4 \) yields the lowest point (-4) compared to others. However, you may also consider that the steepness and shape matter when determining "lower" in comparison to others as well.

In conclusion, without knowing the exact graph details or scales, \( y = \frac{1}{2}|x| - 4 \) is likely to be the correct answer as it extends lower than the others given its vertex at -4.