Question

1. determine each feature of the graph of the function f(x)= \frac{x^2 - 4x}{-x + 4}, if it exists. if the feature does not exist, leave it blank. horizontal asymptote: y = \underline{\space} y-intercept: (0, \underline{\space}) vertical asymptote: x = \underline{\space} x-intercept: (\underline{\space}, 0) hole: (\underline{\space}, \underline{\space})

Explanation:

Step1: Simplify the function

First, factor the numerator and denominator:
The numerator \(x^2 - 4x = x(x - 4)\), and the denominator \(-x + 4 = -(x - 4)\).
So, \(f(x)=\frac{x(x - 4)}{-(x - 4)}\), where \(x
eq4\) (to avoid division by zero).
Canceling out the common factor \((x - 4)\) (for \(x
eq4\)), we get \(f(x)=-x\), with a hole at \(x = 4\) (since the factor \((x - 4)\) was canceled).

Step2: Find Horizontal Asymptote

For a rational function, if the degree of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. But after simplifying, \(f(x)=-x\) (a linear function, not a rational function in the traditional sense with a horizontal asymptote). So, there is no horizontal asymptote.

Step3: Find y - Intercept

To find the y - intercept, set \(x = 0\) in the simplified function \(f(x)=-x\) (we can use the simplified function because the hole is at \(x = 4\), not \(x = 0\)).
\(f(0)=-0 = 0\), so the y - intercept is \((0,0)\).

Step4: Find Vertical Asymptote

After canceling the common factor, the function is a linear function \(f(x)=-x\) (defined for all \(x
eq4\), but the simplified function has no vertical asymptote because the denominator after canceling has no real roots that make it zero. The original denominator had a root at \(x = 4\), but it was canceled, so there is no vertical asymptote.

Step5: Find x - Intercept

To find the x - intercept, set \(f(x)=0\) in the simplified function \(f(x)=-x\).
\(-x = 0\) gives \(x = 0\), so the x - intercept is \((0,0)\).

Step6: Find Hole

The hole occurs at the value of \(x\) where the canceled factor is zero, i.e., \(x - 4=0\) or \(x = 4\).
To find the y - coordinate of the hole, substitute \(x = 4\) into the simplified function \(f(x)=-x\). So, \(f(4)=-4\). Thus, the hole is at \((4,-4)\).

Answer:

Horizontal Asymptote: (no horizontal asymptote, leave blank)
y - Intercept: \((0,0)\)
Vertical Asymptote: (no vertical asymptote, leave blank)
x - Intercept: \((0,0)\)
Hole: \((4,-4)\)