Question

which statements about the graph of the function \\(f(x) = -x^2 - 4x + 2\\) are true? select three options.

the domain is \\(\\{x | x \le -2\\}\\).
the range is \\(\\{y | y \le 6\\}\\).
the function is increasing over the interval \\((-\infty, -2)\\).
the function is decreasing over the interval \\((-4, \infty)\\).
the function has a positive y-intercept.

Explanation:

Find the vertex of the parabola

Using the Quadratic Vertex knowledge point

$$
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$$

Determine the domain and range

Using the Quadratic Range knowledge point

$$
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$$

Analyze intervals of increase and decrease

Using the Quadratic Intervals of Increase knowledge point

$$
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$$

Find the y-intercept

The \(y\)-intercept occurs where \(x = 0\):

$$ f(0) = -(0)^2 - 4(0) + 2 = 2 $$

Since \(2 > 0\), the \(y\)-intercept is positive.

Evaluate the given statements

We compare our findings to the options:

• "The domain is \(\{x \mid x \le -2\}\)": False (domain is all real numbers).
• "The range is \(\{y \mid y \le 6\}\)": True.
• "The function is increasing over the interval \((-\infty, -2)\)": True.
• "The function is decreasing over the interval \((-4, \infty)\)": False (it decreases over \((-2, \infty)\)).
• "The function has a positive y-intercept": True (the \(y\)-intercept is \(2\)).

Answer:

• The domain is \(\{x \mid x \le -2\}\).
• The range is \(\{y \mid y \le 6\}\). (Correct answer)
• The function is increasing over the interval \((-\infty, -2)\). (Correct answer)
• The function is decreasing over the interval \((-4, \infty)\).
• The function has a positive y-intercept. (Correct answer)