Answer:
The true statements about the graph of the function are:
The range of the function is {y : y ≤ 6} ⇒ 2nd
The function is increasing over the interval (–∞ , –2) ⇒ 3rd
The function has a positive y-intercept ⇒ 5th
Explanation:
* Lets explain how to solve the problem
- The function f(x) = -x² - 4x + 2 is a quadratic function represented
graphically by a downward parabola with maximum vertex
∵ The form of the quadratic function is f(x) = ax² + bx + c
∵ The coordinates of the vertex of the parabola are (h , k) where
h = -b/2a and k = f(h)
∵ a = -1 , b = -4
∴ h = -(-4)/2(-1) = 4/-2 = -2
∵ k = f(h)
∴ k = -(-2)² - 4(-2) + 2 = -(4) - (-8) + 2
∴ k = -4 + 8 + 2 = 6
∴ The vertex of the parabola is (-2 , 6)
* Look to the attached figure to find the true statements
∵ The greatest value of the function is 6 ⇒ y-coordinate of the vertex
∵ The range of the function is the values y-coordinates of the points
on the parabola
∴ The range of the function is {y : y ≤ 6}
- The domain of the function is {x : x ∈ R} or (-∞ , ∞)
∵ The value of y increasing after -∞ to the x-coordinate of the vertex
∵ x-coordinate of the vertex is -2
∴ The function is increasing over the interval (–∞ , –2)
- The parabola intersect the y-axis at point (0 , 2)
∵ The y-intercept is the intersection between the parabola and the
y-axis
∵ The parabola intersect the y-axis at point (0 , 2)
∴ The y-intercept is 2
∴ The function has a positive y-intercept