To answer the questions regarding the graph represented by the equation \( x + 2y = 7 \), let's break it down for each part of your inquiry.
1) What is the end behavior of the graph?
The end behavior of the graph refers to what happens as \( x \) approaches large values (both positive and negative). Rearranging the equation into slope-intercept form gives us: \[ y = -\frac{1}{2}x + \frac{7}{2} \] This linear function has a negative slope, indicating that as \( x \) increases, \( y \) decreases. As \( x \) approaches infinity, \( y \) approaches negative infinity.
Response: As \( x \) approaches infinity, \( y \) approaches negative infinity.
2) What would be a reasonable domain for this scenario?
The scenario involves the total amount of chips in a bag over time, which realistically cannot be negative. Thus, the domain extends from the starting point (where time \( x = 0 \)) to when the chips are gone. Considering the equation \( x + 2y = 7 \) suggests that \( y \) must be \( 0 \) at some point.
To find the maximum \( x \): Setting \( y = 0 \): \[ x + 2(0) = 7 \implies x = 7 \] Thus, the reasonable domain is from \( 0 \) to \( 7 \).
Response: Domain: \( [0, 7] \)
3) What would be reasonable outputs for the scenario domain?
The output, which represents the total amount of chips in ounces, also needs to be non-negative. From the equation \( x + 2y = 7 \), we can express \(y\) in terms of \(x\): \[ y = \frac{7 - x}{2} \] For \( x = 0 \), \( y = \frac{7 - 0}{2} = 3.5 \) ounces. For \( x = 7 \), \( y = \frac{7 - 7}{2} = 0 \) ounces. Therefore, as \( x \) varies from \( 0 \) to \( 7 \), \( y \) varies from \( 3.5 \) oz to \( 0 \) oz.
Response: Outputs: \( [0, 3.5] \)
With these responses, you should be able to populate the areas of your request accurately.