Write an equation for a rational function whose graph has all of the indicated features.
x - intercept of -4
y - intercept of -2
vertical asymptote with equation x = 2
horizontal asymptote with equation y = 1
To write the equation of a rational function with these features, we can use the following steps:
Step 1: Begin by identifying the key elements we need to include in our equation: the x-intercept, the y-intercept, the vertical asymptote, and the horizontal asymptote.
Given:
- x-intercept: -4 --> This means that the graph passes through the point (-4, 0).
- y-intercept: -2 --> This means that the graph passes through the point (0, -2).
- Vertical asymptote: x = 2 --> This means that the graph approaches x = 2 as x becomes extremely small or extremely large.
- Horizontal asymptote: y = 1 --> This means that the graph approaches y = 1 as x becomes extremely small or extremely large.
Step 2: Use the given points to represent the numerator and denominator of the rational function.
The x-intercept (-4, 0) means that the numerator has a factor of (x + 4). The y-intercept (0, -2) means that the numerator evaluates to -2 when x = 0. Therefore, the numerator is -2x.
The vertical asymptote x = 2 means that the denominator has a factor of (x - 2). Since we want the vertical asymptote at x = 2, we do not need to include an additional constant.
The horizontal asymptote y = 1 indicates that the degree of the numerator and denominator are the same. So, we have a linear term in both the numerator and denominator.
Step 3: Combine the numerator and denominator to form the equation of the rational function.
The equation of the rational function can be written as:
f(x) = (-2x) / (x - 2)
In summary, the equation of the rational function based on the given features is f(x) = (-2x) / (x - 2).
x - intercept of -4: y = (x+4)
y(0) = 2
so, y - intercept of -2: y = -(x+4)
vertical asymptote at x = 2: y = -(x+4)/(x-2)
y(0) = 2
so, y = (x+4)/(x+2)
that also has a horizontal asymptote at y=1