Unit 4 Graphing Rational Functions Flashcards

How to find the domain of a rational function?

It is all real numbers except any x-values that causes the denominator to equal to 0 (VA or x-values of holes)

How to find the range of a rational function?

Must graph to figure out :(

(Check Day 2 Graphing HW)

How to find the x-intercepts of a rational function?

Set numerator equal to 0 and solve (or if there's a constant at the end set y=0 and solve)

If the numerator can be factored, then there are multiple x-intercepts

How to find the y-intercept of a rational function?

Plug 0 into all x-variables and solve

How to find vertical asymptote(s) in a rational function?

Set denominator equal to 0 and solve (sometimes need to factor)

How to find horizontal asymptote in a rational function?

If equation has just a number in numerator, then HA is the constant at the end. If no constant, then HA=0

If equation is in form p(x)/q(x) with x-variables on top and bottom, then compare the degrees of numerator and denominator.

For a rational in form p(x)/q(x), what is the horizontal asymptote if the numerator degree is greater than the denominator degree?

There is no horizontal asymptote

For a rational in form p(x)/q(x), what is the horizontal asymptote if the denominator degree is greater than the numerator degree?

The horizontal asymptote is at y = 0

For a rational in form p(x)/q(x), what is the horizontal asymptote if the numerator and denominator degrees are the same?

The horizontal asymptote is the quotient of the leading coefficients

How do you find the end behavior of a rational function?

The end behavior is the SAME as the horizontal asymptote

lim(x->∞) f(x) = HA

lim(x->-∞) f(x) = HA

How to find a hole in a rational function?

When a rational function has the same factors on numerator and denominator (crossed out terms). Set the crossed out factor equal to 0 and solve for x.

The y-value of the hole is found by plugging the x-value back into the simplified form.

Which asymptote(s) can NEVER be crossed?

The vertical asymptote can never be crossed

Which asymptote(s) CAN be crossed?

The horizontal asymptote can be crossed.

What should you find using the ORIGINAL rational equation?

Domain and x-value of holes

What should you find using the SIMPLIFIED rational equation?

All of the asymptotes, both of the intercepts, and the y-value of any holes

If a rational equation has ONE horizontal and ONE vertical asymptote, how many branches are there on the graph?

Two

If a rational equation has ONE horizontal and TWO vertical asymptotes, how many branches are there on the graph?

Three

If a rational equation has ONE horizontal and TWO vertical asymptote, how do you find the 'vertex' of the parabola-like branch in the middle of the graph? How does this relate to the range of the graph?

The center of the graph is the midpoint between the two vertical asymptotes.

Plug the midpoint back into the equation to find the y-value of the vertex.

This y-value is included in the range, expressed as an inequality to represent the middle branch of the graph.

How can you sketch a rational graph?

Sketch dashed lines for the asymptotes and open circles for holes.

Plot the intercepts, if any.

Find the 'vertex' of the parabola part of the graph, if any.

Sketch the graph, using reflection or symmetry if needed.

How do you write an equation for a rational graph with a hole?

Write the equation for the graph that you see first. Know how to write basic equations for linear, quadratic, or rational.

If a is the x-value of the hole, and write (x - a) top and bottom of the rational equation.