Rational Function Graphs

A rational function is a function that can be expressed as the ratio of two polynomials, typically in the form $$f(x) = \frac{P(x)}{Q(x)}$$ where P and Q are polynomials.

The y-intercept of a function is the point where the graph of the function intersects the y-axis, which occurs when $$x = 0$$ .

To find the y-intercept, substitute $$x = 0$$ into the function: $$f(0) = -\frac{4}{0-3} = \frac{4}{3}$$ .

A vertical asymptote is a vertical line $$x = a$$ where a function approaches infinity or negative infinity as the input approaches a.

Vertical asymptotes occur at values of $$x$$ that make the denominator zero, provided that the numerator is not also zero at those points.

A hole occurs in the graph of a rational function at a point where both the numerator and denominator are zero, indicating a removable discontinuity.

The vertical asymptote is at $$x = 2$$ .

The x-intercept of a function is the point where the graph intersects the x-axis, which occurs when $$f(x) = 0$$ .

Set the function equal to zero: $$-\frac{4}{x-3} = 0$$ . This gives no solution, indicating there is no x-intercept.

The degree of the numerator and denominator helps determine the end behavior of the function and the presence of horizontal asymptotes.