Key Equations
| Rational Function | [latex]f\left(x\right)=\dfrac{P\left(x\right)}{Q\left(x\right)}=\dfrac{{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[/latex] |
Key Concepts
- We can use arrow notation to describe local behavior and end behavior of the toolkit functions [latex]f\left(x\right)=\frac{1}{x}[/latex] and [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex].
- 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 [latex]x={x}_{1},{x}_{2},\dots ,{x}_{n}[/latex], vertical asymptotes at [latex]x={v}_{1},{v}_{2},\dots ,{v}_{m}[/latex], and no [latex]{x}_{i}=\text{any }{v}_{j}[/latex], then the function can be written in the form [latex]f\left(x\right)=a\dfrac{{\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}}}[/latex]
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 [latex]y=b[/latex] 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 [latex]x=a[/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[/latex]
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