Question

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
y equals Start Absolute Value x End Absolute Value minus 4

y=|x−4|−1

Answer 1

To determine which absolute value equation represents the lower function in a given graph, we should analyze the equations provided:

1. **y = 2|x| - 2**
- This is a V-shaped graph that opens upwards with a vertex at (0, -2). The slope on either side of the vertex is 2.

2. **y = (1/2)|x| - 4**
- This graph also opens upwards with a vertex at (0, -4). The slope on either side of the vertex is 1/2, making it wider compared to the first equation.

3. **y = |x| - 4**
- This graph opens upwards with a vertex at (0, -4). The slope on either side of the vertex is 1, which is steeper than the second equation but less steep than the first.

4. **y = |x - 4| - 1**
- This graph opens upwards with a vertex at (4, -1).

Now, we can evaluate the y-values at x = 0 for each of the equations:

- For **y = 2|x| - 2**: y(0) = 2|0| - 2 = -2
- For **y = (1/2)|x| - 4**: y(0) = (1/2)|0| - 4 = -4
- For **y = |x| - 4**: y(0) = |0| - 4 = -4
- For **y = |x - 4| - 1**: y(0) = |0 - 4| - 1 = 4 - 1 = 3

From the calculated values at x = 0, the lowest y-value is -4, which occurs for both the equations **y = (1/2)|x| - 4** and **y = |x| - 4**.

Since we need to choose which of these two will represent the lower function overall, we typically assume the wider V-shape (in this case, the one with the smaller slope) remains visually lower over larger x-values. Thus, we conclude that:

**Response**: y = (1/2)|x| - 4 is likely the lower function overall in the graph.