a) an equation with a vertical asymptote at x=2; horizontal asymptote at y=0; no x intercept;y intercept is 3
b)a vertical asymptote at x=1; an oblique asymptote at y=2x-1
(a) y = -6/(x-2)
(b) y = 2x-1 + 1/(x-1) = (2x^2-3x+2)/(x-1)
a) To create an equation with a vertical asymptote at x = 2, we can use the rational function form and include the factor (x - 2) in the denominator. Since we want a horizontal asymptote at y = 0, we should have a constant in the numerator. To ensure no x-intercepts, we can exclude any factors in the numerator that would make it equal to zero. Lastly, we want the y-intercept to be 3, so the numerator can be set to 3. Putting these requirements together, the equation becomes:
y = 3 / (x - 2)
b) To create an equation with a vertical asymptote at x = 1 and an oblique asymptote at y = 2x - 1, we can use a rational function form with a linear factor in the denominator. The linear factor should be (x - 1) to create the vertical asymptote at x = 1. Furthermore, the numerator should be a function of x that gets closer to y = 2x - 1 as x approaches positive or negative infinity. A linear function with the same slope, 2, seems appropriate. We can also add a constant, -1, to ensure the oblique asymptote follows the equation. Combining these requirements, the equation becomes:
y = (2x - 1) / (x - 1)
a) To find an equation with a vertical asymptote at x=2, a horizontal asymptote at y=0, and a y-intercept of 3, you can start by considering the characteristics of a rational function. A rational function is a fraction in the form of f(x) = (P(x))/(Q(x)), where P(x) and Q(x) are polynomial functions.
For a vertical asymptote at x=2, the denominator Q(x) should have a factor of (x-2) since we want the function to approach infinity as x gets close to 2.
To have a horizontal asymptote at y=0, the degree of the numerator P(x) should be less than or equal to the degree of the denominator Q(x). Since we want y=0 as x approaches infinity, the degree of the numerator should be less than the degree of the denominator.
Since there are no x-intercepts, the numerator should not have any real solutions.
Since the y-intercept is 3, when x=0, we want the function to equal 3. Therefore, the numerator P(x) should be 3.
Putting all these conditions together, we can construct an equation with the given characteristics:
f(x) = 3 / (x-2)
b) To find an equation with a vertical asymptote at x=1 and an oblique asymptote at y=2x-1, we can use the characteristics of a rational function again.
For a vertical asymptote at x=1, the denominator Q(x) should have a factor of (x-1), ensuring the function approaches infinity as x gets close to 1.
For an oblique asymptote, the degrees of the numerator P(x) and denominator Q(x) should be equal to allow for an oblique line. In this case, we can let both P(x) and Q(x) have a degree of 1.
Putting this together, we can construct an equation with the given characteristics:
f(x) = (2x - 1) / (x - 1)