algebra weeks 3-5

What does the degree of a polynomial tell you about the graph?

It tells you the number of turning points (at most) and the end behavior.

How does the leading coefficient affect the graph of a polynomial?

It determines the direction the graph opens at the ends (up/down)

What is the "end behavior" of a polynomial?

The behavior of the graph as x→±∞x→±∞, determined by the degree and leading coefficient.

What is the multiplicity of a zero?

The number of times a zero is repeated; it’s the exponent on its factor.

What happens if a zero has even multiplicity?

The graph touches the x-axis but does not cross it.

What happens if a zero has odd multiplicity?

The graph crosses the x-axis at that point.

What does it mean if (x−a)k(x−a)k is a factor of a polynomial?

X=a is a root with multiplicity k

What does the Intermediate Value Theorem say about polynomial graphs?

If f(a)f(a) and f(b)f(b) have opposite signs, then there is at least one root between aa and bb.

What is a rational function?

A function that is the ratio of two polynomials: f(x)=N(x)/d(x)

What values are excluded from the domain of a rational function?

Any values that make the denominator zero.

How do you find vertical asymptotes of a rational function?

Set the simplified denominator equal to 0.

How do you find horizontal asymptotes?

Compare degrees of the numerator and denominator:

• If degree top < bottom → y=0y=0

• If degrees equal → ratio of leading coefficients

• If degree top > bottom → no horizontal asymptote (look for slant)

How do you find the x-intercept of a rational function?

Set the numerator equal to 0 and solve.

How do you find the y-intercept of a rational function?

Evaluate f(0)f(0) if 0 is in the domain.

What are the steps to graph a rational function?

1. Determine the domain

2. Identify vertical and horizontal asymptotes

3. Find x- and y-intercepts

4. Plot key points and asymptotes

5. Sketch curves based on sign and shape