Questions: Graph the rational function. g(x) = (-2x + 8) / (x^2 - 6x + 8) Start by drawing the asymptotes (if there are any). Then plot a hollow dot wherever there is a "hole" in the graph.

Graph the rational function.
g(x) = (-2x + 8) / (x^2 - 6x + 8)

Start by drawing the asymptotes (if there are any). Then plot a hollow dot wherever there is a "hole" in the graph.

Transcript text: page - MATH-1130-S33 - College Algebra - MATH A... ALEKS - Sienna Angelo Polynomial and Rational Functions Graphing rational functions with holes Graph the rational function. \[ g(x)=\frac{-2 x+8}{x^{2}-6 x+8} \] Start by drawing the asymptotes (if there are any). Then plot tw a hollow dot wherever there is a "hole" in the graph. Explanation Check

Solution

Solution Steps

Step 1: Factor the numerator and the denominator

The given function is \( g(x) = \frac{-2x + 8}{x^2 - 6x + 8} \).

First, factor the numerator and the denominator:

Numerator: \(-2x + 8 = -2(x - 4)\)
Denominator: \(x^2 - 6x + 8 = (x - 2)(x - 4)\)

So, the function can be rewritten as: \[ g(x) = \frac{-2(x - 4)}{(x - 2)(x - 4)} \]

Step 2: Simplify the function and identify the hole

Simplify the function by canceling the common factor \((x - 4)\): \[ g(x) = \frac{-2}{x - 2} \quad \text{for} \quad x \neq 4 \]

Since \(x = 4\) was canceled out, there is a hole in the graph at \(x = 4\).

Step 3: Identify the vertical asymptote

The simplified function is \( g(x) = \frac{-2}{x - 2} \).

The vertical asymptote occurs where the denominator is zero: \[ x - 2 = 0 \implies x = 2 \]

Final Answer

The hole in the graph is at \( (4, g(4)) \). To find \( g(4) \), substitute \( x = 4 \) into the simplified function: \[ g(4) = \frac{-2}{4 - 2} = \frac{-2}{2} = -1 \] So, the hole is at \( (4, -1) \).
The vertical asymptote is at \( x = 2 \).

Graph the function with a vertical asymptote at \( x = 2 \) and a hole at \( (4, -1) \).

Was this solution helpful?