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]$\left(-4, \infty\right)$[/tex]).
The function has a positive [tex]$y$[/tex]-intercept.
Answer (1)
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 downward, the range is y ∣ y ≤ 6 .
Determine the increasing interval: The function is increasing on the interval ( − ∞ , − 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 ) , and 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 the function to verify 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 plugging x = − 2 into the function: f ( − 2 ) = − ( − 2 ) 2 − 4 ( − 2 ) + 2 = − 4 + 8 + 2 = 6 . So, the vertex is ( − 2 , 6 ) .
Determining the Domain The domain of the function is all real numbers since it is a polynomial. The statement "The domain is x ∣ x ≤ − 2 " is false.
Determining the Range Since the coefficient of the x 2 term is negative ( a = − 1 ), the parabola opens downward. Therefore, the vertex represents the maximum point, and the range is all y values less than or equal to the y-coordinate of the vertex. The range is y ∣ y ≤ 6 . The statement "The range is y ∣ y ≤ 6 " is true.
Determining Increasing Interval For a parabola opening downward, the function is increasing to the left of the vertex and decreasing to the right of the vertex. Since the x-coordinate of the vertex is − 2 , the function is increasing on the interval ( − ∞ , − 2 ) . The statement "The function is increasing over the interval ( − ∞ , − 2 ) " is true.
Determining Decreasing Interval The function is decreasing on the interval ( − 2 , ∞ ) . The statement "The function is decreasing over the interval ( − 4 , ∞ ) " is false because the interval should start at x = − 2 , not x = − 4 .
Determining the 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 The 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 quadratic functions is crucial in many real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By analyzing the vertex and intercepts of the quadratic function representing the projectile's path, they can determine the maximum height reached and the distance traveled. Similarly, in business, quadratic functions can model profit curves, helping companies identify the optimal production level to maximize profits. The vertex of the profit curve indicates the production level at which the maximum profit is achieved. These applications highlight the importance of understanding the properties of quadratic functions in solving practical problems.
