Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. $$ f(x)=x^{2}+10 x+24 $$ (a) The vertex is $$ (-5,-1) $$. (Type an ordered pair, using integers or fractions.) (b) Determine whether the parabola has a maximum value or a minimum value and find the value. Select the correct choice below and fill in the answer box within your choice. (Type an integer or a fraction.) A. The parabola opens downward and has a maximum value of B. The parabola opens upward and has a minimum value of -1 (c) What is the range of $$ f(x) $$ ? The range of $$ f(x) $$ is [ $$ -1, \infty $$ ). (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) (d) On what interval is the function increasing? The function is increasing on $$ [-5, \infty) $$. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.)

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UpStudy AI · Accepted Answer

We are given f(x) = x² + 10x + 24.

Step (a): To find the vertex, use the formula for the x‑coordinate of the vertex, x = –b/(2a).

Here, a = 1 and b = 10, so:
  x = –10/(2·1) = –5.
Now compute f(–5):
  f(–5) = (–5)² + 10(–5) + 24 = 25 – 50 + 24 = –1.
Thus, the vertex is (–5, –1).

Step (b): Since a = 1 (which is positive), the parabola opens upward. This means the vertex represents the minimum point. Therefore, the function has a minimum value of –1.

Step (c): Because the vertex is the minimum point and the parabola opens upward, the range starts at the minimum value –1 and extends to infinity. Thus, the range is [–1, ∞).

Step (d): The function decreases when x is less than the x‑coordinate of the vertex and increases when x is greater than the x‑coordinate of the vertex. Since the vertex occurs at x = –5, the function is increasing for x ≥ –5. In interval notation, the function is increasing on [–5, ∞).

Summary of Answers:
(a) Vertex: (–5, –1)
(b) The parabola opens upward and has a minimum value of –1.
(c) Range: [–1, ∞)
(d) Function is increasing on: [–5, ∞)
(a) Vertex: (–5, –1) (b) The parabola opens upward and has a minimum value of –1. (c) Range: [–1, ∞) (d) Function is increasing on: [–5, ∞)

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