Question

possible points 25
identify the transformation and vertex of the function $g(x) = -6(x + 4)^2 - 3$ from the parent function $f(x) = x^2$. move the correct answer to each box. not all answers will be used.
the transformation of f(x) to g(x) is a horizontal shift 4 units left , a __ and a __
the graph of g(x) is ____ than the graph of f(x)
the vertex is (-4,-3)

options:
vertical shift 3 units down
vertical shift 3 units up

Explanation:

Step1: Identify vertical shift

For $g(x) = -6(x+4)^2 - 3$ compared to $f(x)=x^2$, the $-3$ outside the square indicates a vertical shift 3 units down.

Step2: Identify vertical stretch/reflection

The coefficient $-6$ means the graph is reflected over the x-axis and vertically stretched by a factor of 6, so it is narrower than $f(x)$.

Step3: Confirm vertex

For vertex form $y=a(x-h)^2+k$, vertex is $(h,k)$. Here $h=-4, k=-3$, so vertex is $(-4,-3)$.

Answer:

The transformation of $f(x)$ to $g(x)$ is a horizontal shift 4 units left, a vertical shift 3 units down, and a vertical stretch by a factor of 6 with reflection over the x-axis.
The graph of $g(x)$ is narrower than the graph of $f(x)$.
The vertex is $(-4,-3)$