Question

1. what is is the equation of the function shown by the green dashed line shown in the graph below? (graph with grid and a function curve)

Explanation:

Step1: Identify function type

The graph has a vertical asymptote, so it is a rational function. It also has a "hole" at $x=1$ (since the graph skips this point, but the shape suggests a factor that cancels) and vertical asymptote at $x=6$. The graph passes through the origin $(0,0)$, so the numerator has a root at $x=0$.

Step2: Form basic rational function

Start with a rational function with numerator having $x$ (for root at 0) and $(x-1)$ (for the hole), denominator having $(x-1)$ (cancels for hole) and $(x-6)$ (vertical asymptote):
$y = \frac{ax(x-1)}{(x-1)(x-6)}$ where $a$ is a constant.
Simplify (for $x
eq 1$):
$y = \frac{ax}{x-6}$

Step3: Solve for constant $a$

Use a point on the graph, e.g., when $x=3$, $y=3$. Substitute into the simplified equation:
$3 = \frac{a(3)}{3-6}$
$3 = \frac{3a}{-3}$
$3 = -a$
$a = -3$

Step4: Write final equation

Substitute $a=-3$ into the simplified function, and note the hole at $x=1$:
$y = \frac{-3x(x-1)}{(x-1)(x-6)}$ or simplified (for $x
eq 1$): $y = \frac{-3x}{x-6}$

Answer:

The equation of the green dashed line is $\boldsymbol{y = \frac{-3x}{x-6}}$ (with a removable discontinuity at $x=1$, written fully as $y = \frac{-3x(x-1)}{(x-1)(x-6)}$ for $x
eq 1, 6$)