Question

use the interactive graph below to sketch a graph of $y = -3\log_{3}(1 - x) + 6$. place the asymptote before placing the two points.

Explanation:

Step1: Find vertical asymptote

For logarithmic functions, the argument must be positive. Set $1-x > 0$, so $x < 1$. The vertical asymptote is $x=1$.

Step2: Find x-intercept (y=0)

Set $y=0$:
$$0 = -3\log_3(1-x) + 6$$
$$3\log_3(1-x) = 6$$
$$\log_3(1-x) = 2$$
$$1-x = 3^2 = 9$$
$$x = 1-9 = -8$$
Point: $(-8, 0)$

Step3: Find point when $x=0$

Substitute $x=0$:
$$y = -3\log_3(1-0) + 6 = -3\log_3(1) + 6$$
Since $\log_3(1)=0$, $y=6$. Point: $(0, 6)$

Step4: Find point when $1-x=3$ (x=-2)

Substitute $x=-2$:
$$y = -3\log_3(3) + 6 = -3(1) + 6 = 3$$
Point: $(-2, 3)$

Answer:

Vertical Asymptote: $x=1$
Key Points: $(-8, 0)$, $(0, 6)$ (or $(-2, 3)$ as an additional point)
The graph increases from left to right, approaches $x=1$ (never crossing it), passes through the identified points, and matches the shape of the transformed logarithmic curve.