Question

determine each feature of the graph of the given function.
$f(x) = \frac{-(x - 1)}{3(x - 1)^4}$
answer attempt 3 out of 5

of horizontal asymptotes: one

$y = \boxed{0}$

of holes: none

of vertical asymptotes: one

$x = \boxed{-1}$

of x-intercepts: none

of y-intercepts: one

$(0, \boxed{\frac{1}{3}})$

Explanation:

Step1: Simplify the Function

First, simplify \( f(x)=\frac{-(x - 1)}{3(x - 1)^4} \) by canceling the common factor \((x - 1)\) (note \(x
eq1\) to avoid division by zero). We get \( f(x)=\frac{-1}{3(x - 1)^3} \) for \(x
eq1\).

Step2: Analyze Vertical Asymptote

To find vertical asymptotes, set the denominator equal to zero: \( 3(x - 1)^3 = 0 \) when \(x - 1 = 0\), so \(x = 1\). The previous answer had \(x=-1\), which was incorrect.

Step3: Analyze x - intercepts

To find x - intercepts, set \(y = 0\), i.e., \( \frac{-1}{3(x - 1)^3}=0 \). But a fraction is zero only when the numerator is zero, and the numerator here is \(-1
eq0\), so there are no x - intercepts (this part was correct before).

Step4: Analyze Horizontal Asymptote

For horizontal asymptotes, we look at the degrees of the numerator and denominator. The numerator has degree \(0\) (since it's a constant \(-1\)) and the denominator has degree \(3\). When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\) (this part was correct).

Step5: Analyze y - intercept

To find the y - intercept, set \(x = 0\) in the simplified function \( f(0)=\frac{-1}{3(0 - 1)^3}=\frac{-1}{3(-1)^3}=\frac{-1}{-3}=\frac{1}{3} \). So the y - intercept is \((0,\frac{1}{3})\) (this part was correct).

Step6: Analyze Holes

A hole occurs when there is a common factor in the numerator and denominator that can be canceled. We canceled \((x - 1)\) from the original function, but since \(x = 1\) makes the original denominator zero and the simplified function also has a vertical asymptote at \(x = 1\), there are no holes (this part was correct).

Answer:

• # of Horizontal Asymptotes: One, \( y = 0 \) (correct)
• # of Holes: None (correct)
• # of Vertical Asymptotes: One, \( x = 1 \) (corrected from \(x=-1\))
• # of x - intercepts: None (correct)
• # of y - intercepts: One, \( (0,\frac{1}{3}) \) (correct)