Question

1. list the sequence of steps required to graph the function ( f(x) = -(x + 4)^2 - 6 )

a. horizontal translation 4 units to the right, vertical compression by a factor of 1, vertical translation 6 units down
b. horizontal translation 4 units to the right, reflection in x - axis, vertical translation 6 units down
c. horizontal translation 4 units to the left, vertical translation 6 units up, reflection in x - axis
d. horizontal translation 4 units to the left, reflection in x - axis, vertical translation 6 units down

Explanation:

Step1: Identify parent function

The parent function is $g(x) = x^2$.

Step2: Horizontal translation

For $f(x) = -(x+4)^2 -6$, $x \to x+4$ means shift left 4 units: $g_1(x) = (x+4)^2$.

Step3: Reflect over x-axis

Multiply by -1: $g_2(x) = -(x+4)^2$.

Step4: Vertical translation

Subtract 6, shift down 6 units: $f(x) = -(x+4)^2 -6$.

Step5: Match to options

Compare steps to given choices; option c matches.

Answer:

c. horizontal translation 4 units to the left, vertical translation 6 units up, reflection in x-axis