Question

Find the vertex form of the function. Then find each of the following.

(A) Intercepts
(B) Vertex
(C) Maximum or minimum
(D) Range s(x)=−5x²−10x−4
s(x)= (Type your answer in vertex form.)

Answer 1

The vertex form of the function s(x)=-5x²-10x-4 is s(x) = -5(x + 1)² + 1. The vertex is (-1, 1), which represents a maximum point. The range is y ≤ 1, and there are no x-intercepts.

Step-by-step explanation:

To convert the quadratic function s(x) = -5x² - 10x - 4 into vertex form, we need to complete the square. The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

First, factor out the coefficient of the x² term from the first two terms:

s(x) = -5(x² + 2x) - 4

Next, find the number that will complete the square (b/2)² for the x terms inside the parentheses, where b is the coefficient of x:

(2/2)² = 1² = 1

Add and subtract this number inside the parentheses:

s(x) = -5(x² + 2x + 1 - 1) - 4

Combine terms outside the completed square:

s(x) = -5(x² + 2x + 1) + 5 - 4

s(x) = -5(x + 1)² + 1

This is the vertex form of the function, with (-1, 1) as the vertex. Because the coefficient of the x² term is negative, the parabola opens downwards, and thus the vertex represents a maximum point. The y-intercept is the value of s(x) when x = 0, which is -4. Since the function is a downward-opening parabola, there are no x-intercepts because the discriminant b² - 4ac is negative. The range of the function is y ≤ 1, because the maximum value at the vertex is 1 and the function decreases from there.