Question

Write the equation of the rational function with the following characteristics:

Vertical Asymptote (VA): x = -7
Horizontal Asymptote (HA): y = 4
x-intercept: (0, 0)

Answer 1

The equation of the rational function is
`\( f(x) = (4x(x + 7))/((x + 7)) \).`

Step-by-step explanation:

To construct a rational function with the given characteristics, we can use the information provided about vertical and horizontal asymptotes, as well as the x-intercept.

1. Vertical Asymptote (VA): \( x = -7 \) indicates that the function has a vertical asymptote at \( x = -7 \).

2. Horizontal Asymptote (HA): \( y = 4 \) signifies that the function has a horizontal asymptote at \( y = 4 \).

3. x-intercept: The x-intercept at (0, 0) implies that the function crosses the x-axis at the point (0, 0).

The equation of a rational function with a vertical asymptote at
`\( x = -7 \) is typically expressed as \( f(x) = (P(x))/(Q(x)) \), where the denominator \( Q(x) \) has a factor of \( (x + 7) \) to create the vertical asymptote at \( x = -7 \).`

To satisfy the horizontal asymptote condition at \( y = 4 \), the numerator \( P(x) \) should have the same degree as the denominator. By including \( x \) in the numerator (to meet the x-intercept condition), the equation becomes
`\( f(x) = (4x(x + 7))/((x + 7)) \).`

This function has a vertical asymptote at \( x = -7 \), a horizontal asymptote at \( y = 4 \), and an x-intercept at (0, 0), meeting all the specified characteristics.