Question

The function f(x) = –x3 + 2x2 – x + 5 is graphed on a coordinate grid. Which statements accurately describe the end behavior of the graph of the function?

Answer 1

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

Step-by-step 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