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]$\left\{x \mid x \leq-2\right\}$[/tex].
The range is [tex]$\left\{y \mid y \leq 6\right\}$[/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.

Find the vertex of the parabola: x = − 2 a b ​ = − 2 , f ( − 2 ) = 6 . The vertex is ( − 2 , 6 ) .
Determine the range: Since the parabola opens downwards, the range is y ≤ 6 .
Determine the increasing interval: The function is increasing on ( − ∞ , − 2 ) .
Determine the y-intercept: f ( 0 ) = 2 , which is positive. Thus, the final answer is: The range is y ∣ y ≤ 6 , The function is increasing over the interval ( − ∞ , − 2 ) , The function has a positive y -Intercept.

Explanation

Analyzing the Problem 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. Let's analyze each statement.

Finding the Vertex - x-coordinate First, let's find the vertex of the parabola. 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 .

Finding the Vertex - y-coordinate Now, let's find the y-coordinate of the vertex by substituting x = − 2 into the function: f ( − 2 ) = − ( − 2 ) 2 − 4 ( − 2 ) + 2 = − 4 + 8 + 2 = 6 . So, the vertex is at ( − 2 , 6 ) .

Checking the Domain The domain of a quadratic function is all real numbers, so the statement "The domain is x ∣ x ≤ − 2 " is false.

Checking the Range Since the coefficient of the x 2 term is negative ( a = − 1 ), the parabola opens downwards. This means the vertex represents the maximum point of the function. Therefore, the range is all y values less than or equal to the y-coordinate of the vertex, which is 6. So, the range is y ∣ y ≤ 6 . The statement "The range is y ∣ y ≤ 6 " is true.

Checking Increasing Interval Since the parabola opens downwards, the function is increasing to the left of the vertex and decreasing to the right of the vertex. The vertex is at x = − 2 , so the function is increasing on the interval ( − ∞ , − 2 ) . The statement "The function is increasing over the interval ( − ∞ , − 2 ) " is true.

Checking Decreasing Interval The function is decreasing on the interval ( − 2 , ∞ ) . The statement "The function is decreasing over the interval ( − 4 , ∞ ) " is false, since the function starts decreasing from x = − 2 , not x = − 4 .

Checking y-intercept To find the y-intercept, we set x = 0 in the function: f ( 0 ) = − ( 0 ) 2 − 4 ( 0 ) + 2 = 2 . Since the y-intercept is 2, which is positive, the statement "The function has a positive y -Intercept" is true.

Final Answer Therefore, the three true statements are:



The range is y ∣ y ≤ 6 .
The function is increasing over the interval ( − ∞ , − 2 ) .
The function has a positive y -Intercept.

Examples
Understanding the properties of quadratic functions, such as finding the vertex and intercepts, is crucial in various real-world applications. For example, engineers use this knowledge to design parabolic reflectors for satellite dishes or solar cookers, optimizing the focus of incoming signals or sunlight. Similarly, in projectile motion, the path of an object thrown into the air follows a parabolic trajectory, and understanding the vertex helps determine the maximum height the object will reach. These applications highlight the importance of mastering quadratic functions for problem-solving in science and engineering.

Answered by GinnyAnswer | 2025-07-08