Which statements about the graph of the function [tex]$f(x)=-x^2-4 x+2$[/tex] are true? Select three options.
The domain is [tex]$\displaystyle {\{x \mid x \leq-2\}}$[/tex].
The range is [tex]$\displaystyle {\{y \mid y \leq 6\}}$[/tex].
The function is increasing over the interval ([tex]$-\infty,-2$[/tex]).
The function is decreasing over the interval ([tex]$-4, \infty$[/tex]).
The function has a positive [tex]$y$[/tex]-intercept.

The domain of the quadratic function is all real numbers, so the first statement is false.
The vertex of the parabola is ( − 2 , 6 ) , and since it opens downwards, the range is y ∣ y ≤ 6 , making the second statement true.
The function is increasing on the interval ( − ∞ , − 2 ) , so the third statement is true.
The function is decreasing on the interval ( − 2 , ∞ ) , making the fourth statement false.
The y-intercept is 2, which is positive, so the fifth statement is true.

Therefore, the three true statements are: The range is y ∣ y ≤ 6 , The function is increasing over the interval ( − ∞ , − 2 ) , and The function has a positive y -intercept.
Explanation

Analyzing the Statements We are given the quadratic function f ( x ) = − x 2 − 4 x + 2 and asked to determine which of the given statements about its graph are true. We will analyze each statement.

Domain: The domain of a quadratic function is all real numbers, since we can plug in any real number for x . The statement "The domain is x ∣ x ≤ − 2 " is false.

Range: Since the coefficient of the x 2 term is negative ( − 1 ), the parabola opens downwards. This means the function has a maximum value at its vertex. We need to find the vertex to determine the range. The x -coordinate of the vertex is given by x = − 2 a b ​ , where a = − 1 and b = − 4 . Thus, x = − 2 ( − 1 ) − 4 ​ = − − 2 − 4 ​ = − 2 . The y -coordinate of the vertex is f ( − 2 ) = − ( − 2 ) 2 − 4 ( − 2 ) + 2 = − 4 + 8 + 2 = 6 . So the vertex is ( − 2 , 6 ) . Since the parabola opens downwards, the range is all y values less than or equal to 6. The statement "The range is y ∣ y ≤ 6 " is true.

Increasing Interval: For a parabola opening downwards, the function is increasing to the left of the vertex. Since the vertex is at x = − 2 , the function is increasing on the interval ( − ∞ , − 2 ) . The statement "The function is increasing over the interval ( − ∞ , − 2 ) " is true.

Decreasing Interval: For a parabola opening downwards, the function is decreasing to the right of the vertex. Since the vertex is at x = − 2 , the function is decreasing on the interval ( − 2 , ∞ ) . The statement "The function is decreasing over the interval ( − 4 , ∞ ) " is false, because the function starts decreasing from x = − 2 , not x = − 4 .

y-intercept: The y -intercept is the value of the function when x = 0 . So, f ( 0 ) = − ( 0 ) 2 − 4 ( 0 ) + 2 = 2 . Since 0"> 2 > 0 , the function has a positive y -intercept. The statement "The function has a positive y -intercept" is true.


Examples
Understanding the properties of quadratic functions, like the one in this problem, is crucial in many real-world applications. For example, engineers use quadratic functions to model the trajectory of projectiles, such as a ball thrown in the air or the path of a rocket. By knowing the vertex, domain, range, and increasing/decreasing intervals, they can predict the maximum height the projectile will reach, where it will land, and how its speed changes over time. This knowledge is also used in optimization problems, such as finding the dimensions of a garden that maximize the area enclosed by a given amount of fencing.

Answered by GinnyAnswer | 2025-07-05