Question

question 1 (3 points)
graph the function:
$f(x) = 2x^2 + 4x + 4$
determine whether the function is increasing, decreasing, constant, or undefined over the interval $(-infty, -1)$.
\\(\circ\\) a increasing
\\(\circ\\) b decreasing
\\(\circ\\) c constant
\\(\circ\\) d undefined

Explanation:

Step1: Find the vertex of the parabola

The function \( f(x) = 2x^2 + 4x + 4 \) is a quadratic function in the form \( ax^2 + bx + c \). The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = 4 \), so \( x = -\frac{4}{2\times2} = -1 \).

Step2: Determine the direction of the parabola

Since \( a = 2 > 0 \), the parabola opens upwards.

Step3: Analyze the interval \((-\infty, -1)\)

For a parabola that opens upwards, the function is decreasing to the left of the vertex (where the x-coordinate of the vertex is -1) and increasing to the right of the vertex. So, over the interval \((-\infty, -1)\), the function is decreasing.

Answer:

b. Decreasing