Question

1. complete the following.

part a. describe the transformation of $f$ represented by $g$.
$f(x)=0.5^{x}$, $g(x)=6(0.5)^{x+5}-2$
the graph of $g$ is a $square$ by a factor of $square$, followed by a translation $square$ units left and 2 units $square$ of the graph of $f$.

available options:
2, 4, 5, 6
up, down, left, right
vertical shrink, vertical stretch

Explanation:

Step1: Identify vertical stretch factor

For $g(x)=6(0.5)^{x+5}-2$ and $f(x)=0.5^x$, the coefficient 6 outside the exponential term indicates a vertical stretch.

Step2: Identify horizontal translation

The $x+5$ inside the exponent means $x$ is replaced by $x-(-5)$, so this is a left translation by 5 units.

Step3: Identify vertical translation

The $-2$ at the end of $g(x)$ shifts the graph downward by 2 units.

Step4: Match to blanks

Blank 1: vertical stretch; Blank 2: 6; Blank 3: 5; Blank 4: down

Answer:

The graph of $g$ is a vertical stretch by a factor of $\boldsymbol{6}$, followed by a translation $\boldsymbol{5}$ units left and 2 units down of the graph of $f$.