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 $\boldsymbol{horizontal shift 4 units left}$ $\boldsymbol{downarrow}$, a $\boldsymbol{square}$ $\boldsymbol{downarrow}$ and a $\boldsymbol{square}$ $\boldsymbol{downarrow}$.
the graph of g(x) is $\boldsymbol{square}$ $\boldsymbol{downarrow}$ than the graph of f(x).
the vertex is $\boldsymbol{square}$ $\boldsymbol{downarrow}$

Explanation:

Step1: Identify vertical stretch/reflection

For $g(x) = -6(x+4)^2 - 3$ vs $f(x)=x^2$, the coefficient $-6$ means a vertical stretch by factor 6 and reflection over the x-axis.

Step2: Identify vertical shift

The $-3$ at the end means a vertical shift 3 units down.

Step3: Compare graph width/orientation

Since $| -6 | > 1$ and it's negative, the graph is narrower and opens downward (so it's narrower than $f(x)$).

Step4: Find vertex of $g(x)$

Vertex form is $a(x-h)^2 + k$, so $h=-4, k=-3$. Vertex is $(-4, -3)$.

Answer:

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