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Advanced Algebra Chapter 9. Rational Equations and Functions. Inverse and Joint Variation—9.1. Direct Variation. Two variables x and y vary directly iff: If x and y vary directly and y=6 when x=3, write the general direct variation equation. Inverse Variation.
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Advanced Algebra Chapter 9 Rational Equations and Functions
Direct Variation • Two variables x and y vary directly iff: • If x and y vary directly and y=6 when x=3, write the general direct variation equation
Inverse Variation • x and y vary inversely if they are related by: • k is our constant of variation
Direct or Inverse? • How do we tell the difference? • Direct: • Constant and x are multiplied • Inverse: • Constant is divided by x • x is the denominator
Writing Equations • x and y vary inversely, and y=6 and x=1.5 • Find y when
Joint Variation • When a quantity varies directly as the product of two or more other variables • However, there are other possibilities of joint variation
Variation y varies directly with x Y varies inversely with x
Variation z varies jointly with x and y y varies inversely with the square of x
Variation z varies directly with y and inversely with x y varies inversely with x and z
Domain and Range • Domain: Any and all numbers that can be plugged into a function • X-values • Range: All output values of a function • Y-values
Rational Functions • Any function composed of the quotient of two functions
Hyperbolas • The x-axis is the horizontal asymptote • The y-axis is the verticalasymptote • Domain: All values except 0 • Range: All values except 0 • Contains two symmetrical parts called branches
Hyperbolas--Shifting • All functions in the form are hyperbolas • Vertical asymptote at: • Horizontal asymptote at:
Other Hyperbolas • All functions of form are also hyperbolas • Vertical asymptote: • Horizontal asymptote:
Graphs of Rational Functions • The graph of the of the functionhas the following: • The x-intercepts of the graph are the real zeros of • The vertical asymptote occurs at each real zero of
Graphs of Rational Functions • The graph of the of the functionhas the following: • The graph has at most 1 horizontal asymptote • If , the line is the hor. asym. • If , the line is the hor. asym. • If , the graph has no hor. asym. The graph’s end behavior is that of the line:
Rational Expressions • A rational expression is in simplest form iff the numerator and denominator share no common factors (other than 1)
Simplifying Expressions • Two Step process • Factor the numerator and denominator completely • Divide out any common factors
Addition and Subtraction • With any fraction… • When adding or subtracting must have a common denominator • Example:
