Question

How do you graph `f(x)=-3/(x-1)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

Holes: `color(green)"Since nothing was cancelled out, there are no holes."`

Vertical Asymptote: Use the denominator. The function is undefined when the denominator is zero. At what value of `x` will the denominator equal zero?

`color(green)"Set the denominator to zero, and then solve for x"`.

`color(green)"x - 1 = 0"`
`color(green)"x = 1"`

`color(green)"The vertical asymptote will be at x = 1"`.

Horizontal Asymptote: Use the degrees of the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, HA is at y=0.
• If the degree of the numerator is more than the degree of the denominator, there is no HA.
• If they are equal, divide the coefficient of `x^(degree)` in the numerator by the coefficient of `x^(degree)` in the denominator.

`color(green)"The degree of the numerator is zero. The degree of the denominator is 1. The HA is at y=0."`

x-int: The graph will pass through the x-axis when y is equal to zero.

`color(green)"Set y = 0, then solve for x."`

`color(green)0 color(green)= - color(green)3/color(green)(x-1)`

`color(green)"In this case, this is not possible, therefore there is no x-intercept."`

y-int: The graph will pass through the y-axis when x is equal to zero.

`color(green)"Set x = 0, then solve for y."`

`color(green)y color(green)= - color(green)3/color(green)(0-1)`

`color(green)y color(green)= color(green)3`

`color(green)"The graph passes through the y-axis at 3."`

Use these values to plot points, then use a table of values if extra points are needed.