Question

which is the range of $y = 3(x + 4)^2 - 30$?
options:
a. $x \leq 2$
b. none of these
c. $y \leq 30$
d. $y \leq 2$
e. $y \geq 0$
f. $y \geq 30$
g. $y \leq -30$
h. $y \leq 4$
i. $y \geq -30$
j. $y \geq -4$
k. $y \leq 0$
l. $x \geq -4$

Explanation:

Step1: Analyze the function form

The given function is \( y = 3(x + 4)^2 - 30 \). This is a quadratic function in vertex form \( y=a(x - h)^2 + k \), where \( a = 3 \), \( h=-4 \), and \( k = - 30 \).

Step2: Determine the direction of the parabola

Since \( a = 3>0 \), the parabola opens upwards.

Step3: Find the minimum value of \( y \)

For a parabola opening upwards, the vertex \((h,k)\) gives the minimum point. Here, the vertex is \((-4,-30)\), so the minimum value of \( y \) is \( - 30 \).

Step4: Determine the range (image) of the function

Since the minimum value of \( y \) is \( - 30 \) and the parabola opens upwards, the range of \( y \) is \( y\geq - 30 \).

Answer:

\( y\geq - 30 \) (corresponding to the option with \( y\geq - 30 \))