1.1 Functions
inverse variation function
f(x) = a/x^n --> ax^-n where a≠0, n>0 special case of power function f(x) varies inversely with x or the nth power of x Both axes are asymptotes. D: x≠0, most real-world applications use x>0
Polynomial function
f(x)=a(n)x^n, a(n-1)x^(n-1)...a(1)x+a(0) where n is a whole number f(x) is a polynomial function of x. n=3 --> cubic, n=4 --> quartic The graph cross the x-axis at most n times and has n-1 vertices (changes direction) Domain: all real numbers Range: depends on the equation
exponential function
f(x)=ab^x where a≠0, b>0, b≠1 --> constant with variable exponent f(x) varies exponentially with x Graph cross the y-axis at a x-axis is the asymptote range can be changed by making a negative and shifting it up and down.
Linear function
f(x)=ax+b where a is the slope f(x) varies linearly with x specific type of polynomial straight line: f(x) changes at a constant rate as x changes domain: all real numbers unless equation is x=t range: all real numbers unless m=0 --> y=t
Quadratic function
f(x)=ax^2+bx+c f(x) varies quadratically with x Specific type of polynomial changes direction at 1 vertex D: All real numbers R: depends on vertex and a factoring, completing the square, quadratic formula Graphing with vertex form: vertex --> over 1 up a --> over 1 up 3a --> over 1 up 5a...
Power function
f(x)=ax^b where a≠0, b≠0 --> variable with constant exponent f(x) varies directly with the bth power of x D: depends on b. If b>0 x=all real numbers. If b<0 (inverse function) x≠0 Most real-world applications: if b>0, x=all real numbers, if b<0, x>0
Direct variation function
f(x)=ax^n Special case of linear, polynomial, or power function f(x) varies directly with x goes through the origin D: all real numbers, most real-life applications use x≥0
Rational algebraic function
f(x)=n(x)/d(x) where n and d are polynomial functions f(x) is a rational function of x Has a discontinuity (asymptote or missing point) where denominator is 0, may have other HAs and VAs Range: everything but HA and hole All inverse power functions are rational functions, not all rational functions are inverse power functions
