Question

determine each feature of the graph of the given function. f(x) = (-2x + 4)/(x² - 2x) answer attempt 1 out of 5 horizontal asymptote: y = no horizontal asymptote vertical asymptote: x = no vertical asymptote x-intercept: ( , 0) no x-intercept y-intercept: (0, ) no y-intercept hole: ( , ) no hole

Explanation:

Step1: Simplify the function

Factor numerator and denominator:
Numerator: \(-2x + 4 = -2(x - 2)\)
Denominator: \(x^2 - 2x = x(x - 2)\)
Simplify (cancel \(x - 2\) where \(x
eq 2\)): \(f(x) = \frac{-2}{x}\) (with restrictions \(x
eq 0, 2\))

Step2: Horizontal Asymptote

For rational functions, compare degrees of numerator (\(n\)) and denominator (\(d\)):

• \(n = 0\) (degree of \(-2\)), \(d = 1\) (degree of \(x\))
• Since \(n < d\), horizontal asymptote is \(y = 0\) (previous answer \(y=1\) was wrong).

Step3: Vertical Asymptote

Denominator of simplified function (or original, after canceling) is \(x\) (from \(f(x) = \frac{-2}{x}\)) and \(x
eq 0\), \(x
eq 2\) (from original). Vertical asymptote at \(x = 0\) (since \(x = 2\) was a hole, not asymptote).

Step4: x-Intercept

Set numerator of simplified function to 0: \(-2 = 0\) (never true) → No x-intercept.

Step5: y-Intercept

Set \(x = 0\), but \(x = 0\) is undefined (denominator 0) → No y-intercept.

Step6: Hole

Hole at \(x = 2\) (canceled factor \(x - 2\)). Plug \(x = 2\) into simplified function: \(f(2) = \frac{-2}{2} = -1\) → Hole at \((2, -1)\).

Answer:

Horizontal Asymptote: \(y = 0\)
Vertical Asymptote: \(x = 0\)
x-Intercept: No x-intercept
y-Intercept: No y-intercept
Hole: \((2, -1)\)