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x + 2y = 7

1) What is the end behavior of the graph? Response area

2) What would be a reasonable domain for this scenario? Response area

3) What would be reasonable outputs for the scenario domain?

1 answer

Let's analyze the equation \(x + 2y = 7\), which can be rewritten in slope-intercept form as \(y = -\frac{1}{2}x + \frac{7}{2}\). This allows us to understand its graph better.

1) End Behavior of the Graph:

The end behavior of a linear function is determined by its slope. For the equation \(y = -\frac{1}{2}x + \frac{7}{2}\):

  • As \(x\) approaches positive infinity \((x \to +\infty)\), \(y\) will approach negative infinity \((y \to -\infty)\) because of the negative slope.
  • As \(x\) approaches negative infinity \((x \to -\infty)\), \(y\) will approach positive infinity \((y \to +\infty)\).

End Behavior Response:

  • As \(x \to +\infty\), \(y \to -\infty\).
  • As \(x \to -\infty\), \(y \to +\infty\).

2) Reasonable Domain for the Scenario:

For the domain of a linear equation, the most common assumption is that \(x\) can take any real number. However, if this equation represents a real-world scenario (like a budget, distance, or height), the domain can be restricted to non-negative values or other logical limits based on context.

Reasonable Domain Response:

A reasonable domain could be \(x \geq 0\) (non-negative values), or you could define it based on specific constraints if the scenario was given.

3) Reasonable Outputs for the Scenario Domain:

To find reasonable outputs (\(y\)) corresponding to the domain, we'll substitute values of \(x\) into the equation \(y = -\frac{1}{2}x + \frac{7}{2}\).

For example:

  • If \(x = 0: y = -\frac{1}{2}(0) + \frac{7}{2} = \frac{7}{2} = 3.5\)
  • If \(x = 2: y = -\frac{1}{2}(2) + \frac{7}{2} = -1 + \frac{7}{2} = \frac{5}{2} = 2.5\)
  • If \(x = 4: y = -\frac{1}{2}(4) + \frac{7}{2} = -2 + \frac{7}{2} = \frac{3}{2} = 1.5\)
  • If \(x = 6: y = -\frac{1}{2}(6) + \frac{7}{2} = -3 + \frac{7}{2} = \frac{1}{2} = 0.5\)

As \(x\) increases further, \(y\) will eventually become negative.

Reasonable Outputs Response:

  • If \(x = 0\), then \(y = 3.5\).
  • If \(x = 2\), then \(y = 2.5\).
  • If \(x = 4\), then \(y = 1.5\).
  • If \(x = 6\), then \(y = 0.5\).
  • If \(x = 8\), then \(y = -0.5\) (indicating \(y\) can take negative values as \(x\) increases).

In response to reasonable outputs based on the domain \(x \geq 0\), the outputs will start from 3.5 and decrease as \(x\) increases.