02 Polynomial and Rational Functions Free Online FlashCards

What is the standard form of a linear function?

A linear function has the form $f(x)=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

What is the axis of symmetry of a quadratic function?

The axis of symmetry of $f(x)=ax^2+bx+c$ is $x=-\frac{b}{2a}$.

How does the discriminant classify quadratic zeros?

For $ax^2+bx+c=0$, $\Delta=b^2-4ac$. If $\Delta>0$, there are two distinct real zeros; if $\Delta=0$, one repeated real zero; if $\Delta<0$, no real zeros.

What does the Factor Theorem state?

The Factor Theorem states that $f(r)=0$ if and only if $(x-r)$ is a factor of $f(x)$.

How does multiplicity affect a polynomial graph?

An odd-multiplicity zero generally makes the graph cross the $x$-axis; an even-multiplicity zero generally makes it touch the axis and turn around.

What candidates does the Rational Zero Theorem provide?

If $p/q$ is a rational zero in lowest terms, then $p$ divides the constant term and $q$ divides the leading coefficient.

What is the end behavior of an odd-degree polynomial with positive leading coefficient?

An odd-degree polynomial with a positive leading coefficient falls toward $-\infty$ on the left and rises toward $\infty$ on the right.

What is the maximum number of turning points for a degree-$n$ polynomial?

A degree-$n$ polynomial can have at most $n-1$ turning points.

When does a polynomial inequality include its zeros?

For a polynomial inequality, include a real zero when the inequality is $\le$ or $\ge$; exclude it when the inequality is $<$ or $>$.

How do holes differ from vertical asymptotes?

A canceled common factor creates a hole, while a denominator zero that remains after cancellation creates a vertical asymptote.

How is a horizontal asymptote found when degrees are equal?

If the numerator and denominator have equal degree, the horizontal asymptote is the ratio of their leading coefficients.

When does a rational function have a slant asymptote?

A rational function has a slant asymptote when the numerator's degree is exactly one greater than the denominator's degree. Polynomial division finds the asymptote.