Analyze the function algebraically. List its vertical asymptotes, holes, y-intercept, and horizontal asymptote, if any.
q(x)= -4/x-2
vertical asymptote=
hole=
horizontal asymptote=
y-intercept=
To analyze the function algebraically, we need to examine its expression and identify any restrictions or characteristics.
The given function is q(x) = -4 / (x - 2).
Vertical asymptotes: Vertical asymptotes occur when the denominator of a rational function equals zero and the numerator does not. In this case, the denominator (x - 2) equals zero when x = 2. Therefore, there is a vertical asymptote at x = 2.
Hole: A "hole" in a function occurs when both the numerator and denominator equal zero at a certain point, canceling each other out. In the given function, there is no such cancellation, so there are no holes.
Horizontal asymptote: To determine if there is a horizontal asymptote, we need to compare the degrees of the numerator and the denominator. In this case, the numerator is a constant (-4), and the denominator is linear (x - 2). Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0 (the x-axis).
Y-intercept: To find the y-intercept, we set x = 0 in the function q(x). Plugging in x = 0, we have q(0) = -4 / (0 - 2) = -4 / (-2) = 2. Therefore, the y-intercept is (0, 2).
Summary:
- Vertical asymptote: x = 2
- Hole: None
- Horizontal asymptote: y = 0 (x-axis)
- Y-intercept: (0, 2)