Given the following features of a rational function:x-intercept: -2 y-intercept: -3 vertical asymptote: x = 2horizontal asymptote: y = 3 Which of the following is the correct equation?

Question

Given the following features of a rational function:x-intercept: -2 y-intercept: -3 vertical asymptote: x = 2horizontal asymptote: y = 3 Which of the following is the correct equation?

BlackTom AI · Accepted Answer

To solve this, I’ll go through what each feature implies about the rational function and then evaluate each option.
First, since the x-intercept is -2, the numerator must be zero at x = -2. That means the numerator contains a factor (x + 2).
Next, the y-intercept is -3, so when x = 0, f(0) should equal -3. This provides a numerical check once we choose a form with the (x + 2) factor in the numerator and a suitable denominator.
The vertical asymptote at x = 2 tells us the denominator must have a factor (x − 2). Any valid form must have a (x − 2) in the denominator, not (x + 2).
The horizontal asymptote y = 3 implies that the degrees of the numerator and denominator are the same and the ratio of their leading coefficients is 3. In other words, as x grows large, f(x) approaches 3.
Now examine each option:
- Option a: f(x) = (-3x + 6)/(x − 2). The numerator equals -3(x − 2), which has a zero at x = 2, not at x = -2. That would place the x-intercept at 2, conflicting with the given x-intercept. Also, the intercepts don’t align with the required sign and value.
- Option b: f(x) = (3x + 6)/(x + 2). Here the denominator is x + 2, which would create a vertical asymptote at x = -2, not x = 2. This contradicts the given asymptote and the x-intercept structure since the numerator vanishes at x = -2, not influencing the vertical asymptote location correctly.
- Option c: f(x) = (3x + 6)/(x − 2) = 3(x + 2)/(x − 2). The numerator has a zero at x = -2, giving the required x-intercept. The denominator gives a vertical asymptote at x = 2. The leading terms are 3x in the numerator and x in the denominator, so the horizontal asymptote is y = 3, matching the provided horizontal asymptote. Finally, evaluating the y-intercept: f(0) = (0 + 6)/(0 − 2) = 6/(-2) = -3, which matches the given y-intercept.
- Option d: f(x) = (3x − 6)/(x + 2) = 3(x − 2)/(x + 2). The zero of the numerator is x = 2, which would place the x-intercept at 2 rather than -2. The denominator x + 2 gives a vertical asymptote at x = -2, which is inconsistent with the specified x = 2. The horizontal asymptote would still be y = 3, but the intercepts and asymptote locations do not align with the data.
In summary, options a, b, and d fail to satisfy at least one of the given conditions (x-intercept, vertical asymptote, y-intercept), whereas option c satisfies all of them: the intercept at x = -2, the y-intercept of -3, the vertical asymptote at x = 2, and the horizontal asymptote at y = 3. The clean way to see the consistency is to check that the numerator vanishes at -2, the denominator vanishes at 2, and the ratio of leading coefficients yields the horizontal asymptote y = 3, all of which are met by option c.

MHF4U - Advanced Functions 12 (2025-26) - A

Given the following features of a rational function:x-intercept: -2 y-intercept: -3 vertical asymptote: x = 2horizontal asymptote: y = 3 Which of the following is the correct equation?

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