Ask a New Question
Menu

The graph below represents the total amount of chips in the bag in oz., y, over time, x, in minutes. The image shows a graph of the linear equation x + 2y = 7 on a coordinate plane. The line passes through the points (0, 3.5) and (7, 0). The x-axis and y-axis are labeled, and grid lines are present to show the scale. The equation of the line is also labeled on the graph. The graph is titled "Figure 1."

1) What is the end behavior of the graph?

2) What would be a reasonable domain (input) for this scenario?

3) What would be reasonable range (outputs) for the scenario?

End Behavior responces:
As x goes to - ∞, y goes to + ∞. As x goes to + ∞, y goes to - ∞.
As x goes to - ∞, y goes to - ∞. As x goes to + ∞, y goes to ∞.

Reasonable domain:
[0, 7]
[0, 3.5]

Reasonable outputs:
[0, 3.5]
[0, 7]
[0, 8]

1 answer

Let's analyze the linear equation given in the context of the scenario.

  1. End Behavior of the Graph: Since the equation of the line is \( x + 2y = 7 \), we can rewrite it in slope-intercept form (y = mx + b):

\[ 2y = -x + 7 \quad \Rightarrow \quad y = -\frac{1}{2}x + \frac{7}{2} \]

The slope is -1/2, indicating that as \( x \) increases (goes to +∞), \( y \) decreases (goes to -∞). Conversely, as \( x \) decreases (goes to -∞), \( y \) increases (goes to +∞). Thus, the end behavior is:

As \( x \) goes to -∞, \( y \) goes to +∞. As \( x \) goes to +∞, \( y \) goes to -∞.

  1. Reasonable Domain (Input): In the context of this scenario (chips in a bag over time), \( x \) represents time in minutes and \( y \) represents the amount of chips in ounces. Since you cannot have negative time or more than the total amount of chips in the bag, a reasonable domain is:

[0, 7] (since 7 minutes is the maximum time based on the graph).

  1. Reasonable Range (Outputs): The output \( y \) (the total amount of chips in ounces) changes as \( x \) varies. The maximum amount of chips occurs at when \( x = 0 \) (which is 3.5 ounces). The minimum occurs when \( x = 7 \) (which is 0 ounces). Therefore, a reasonable range for the scenario is:

[0, 3.5] (since 0 oz is the minimum and 3.5 oz is the maximum according to the graph).

Summary:

  1. End Behavior: As \( x \) goes to -∞, \( y \) goes to +∞. As \( x \) goes to +∞, \( y \) goes to -∞.
  2. Domain: [0, 7]
  3. Range: [0, 3.5]