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Study Guides > MATH 1314: College Algebra

Key Concepts & Glossary

Key Equations

Rational Function f(x)=P(x)Q(x)=apxp+ap1xp1+...+a1x+a0bqxq+bq1xq1+...+b1x+b0,Q(x)0f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}=\frac{{a}_{p}{x}^{p}+{a}_{p - 1}{x}^{p - 1}+...+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q - 1}{x}^{q - 1}+...+{b}_{1}x+{b}_{0}}, Q\left(x\right)\ne 0

Key Concepts

  • We can use arrow notation to describe local behavior and end behavior of the toolkit functions f(x)=1xf\left(x\right)=\frac{1}{x}\\ and f(x)=1x2f\left(x\right)=\frac{1}{{x}^{2}}\\.
  • A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one vertical asymptote.
  • Application problems involving rates and concentrations often involve rational functions.
  • The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.
  • The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is not zero.
  • A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and denominator to be zero.
  • A rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions.
  • Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior.
  • If a rational function has x-intercepts at x=x1,x2,,xnx={x}_{1},{x}_{2},\dots ,{x}_{n}\\, vertical asymptotes at x=v1,v2,,vmx={v}_{1},{v}_{2},\dots ,{v}_{m}, and no xi=any vj{x}_{i}=\text{any }{v}_{j}\\, then the function can be written in the form f(x)=a(xx1)p1(xx2)p2(xxn)pn(xv1)q1(xv2)q2(xvm)qnf\left(x\right)=a\frac{{\left(x-{x}_{1}\right)}^{{p}_{1}}{\left(x-{x}_{2}\right)}^{{p}_{2}}\cdots {\left(x-{x}_{n}\right)}^{{p}_{n}}}{{\left(x-{v}_{1}\right)}^{{q}_{1}}{\left(x-{v}_{2}\right)}^{{q}_{2}}\cdots {\left(x-{v}_{m}\right)}^{{q}_{n}}}\\

Glossary

arrow notation
a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value
horizontal asymptote
a horizontal line = b where the graph approaches the line as the inputs increase or decrease without bound.
rational function
a function that can be written as the ratio of two polynomials
removable discontinuity
a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function
vertical asymptote
a vertical line = a where the graph tends toward positive or negative infinity as the inputs approach a

Licenses & Attributions