Question

question 2 (3 points)
graph the function:
$f(x) = 2x^2 + 4x + 4$
then determine whether the function is increasing, decreasing, constant, or undefined over the interval $(1, \infty)$.
\\(\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 of a quadratic function \( y = ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \). Here, \( a = 2 \), \( b = 4 \), 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 \((1,\infty)\)

For a parabola that opens upwards, the function is increasing to the right of the vertex. The vertex is at \( x=-1 \), and the interval \((1,\infty)\) is to the right of \( x = - 1 \). So, for the interval \((1,\infty)\), as \( x\) increases, \( f(x)\) increases.

Answer:

a. Increasing