Study Flashcards
A rational function has the form
f(x) = N(x)/D(x) where N(x) and D(x) are polynomials
f(x) = N(x)/D(x) where N(x) and D(x) are polynomials
Rational number form
A number is NOT a polynomial when
Negative exponents, variable in denominator, or radicals
Negative exponents, variable in denominator, or radicals
Factors that make a number NOT a polynomial
How is the domain of a number with 4 and 2 as its zeros written?
(-∞, 2)U(2,4)U(4, ∞) because 3 is still included
When a line has an excluded number from the domain, what happens to the graph?
A hole in the line
A hole in a line on a graph appears when
A function has an excluded number from the domain at a certain x
How is f(x)=1/x drawn on graph?
Day 16 page 3
If x > 0, as x gets smaller (close to 0), 1/x gets larger without bound. How is this represented?
Lim x — > 0 +
Lim x — > 0 +
As x approaches 0, the function gets bigger with bound
If x < 0, as x gets closer to 0, 1/x gets smaller without bound. How is this represented?
Lim x — > 0 -
Lim x — > 0 -
As x approaches 0, the function gets smaller with bound
The line x = a is a vertical asymptote of f if
Lim f(x) = +- ∞ x — > a
Lim f(x) = +- ∞ x — > a
The line x = a is a vertical asymptote of f
As x gets larger (x > 0), or farther from 0, 1/x gets smaller approaching 0. How is this represented?
Lim 1/x = 0
x — > ∞ and x — > -∞
Lim 1/x = 0
x — > ∞ and x — > -∞
As x gets larger (x > 0), or farther from 0, 1/x gets smaller approaching 0.
The line y=k is a horizontal asymptote of f if
Lim f(x) = k x — > +- ∞
Lim f(x) = k x — > +- ∞
The line y=k is a horizontal asymptote of f
The function g(x) = ax + b/ cx +d is a transformation of
f(x) = 1/x
Transformation of 1/x
g(x) = ax + b/ cx +d
What transformations occur in:
g(x) = -2 / x+1
Reflection over x-axis, shift left 1, stretch vertical 2
How do you find the transformations of:
h(x) = 3x-2 / x-1
Use long division. The quotient is added and the remainder is the numerator
If f(x) = N(x) / D(x) where N(x) and D(x) have no common factors, then x=a is a vertical asymptote of f iff
D(a) = 0
D(a) = 0
If f(x) = N(x) / D(x) where N(x) and D(x) have no common factors, then x=a is a vertical asymptote of f
What are the vertical asymptotes of g(x) = 1 / x^2 -9?
x = +- 3
How do you find the vertex of a quadratic?
(-b / 2a, f(-b / 2a))
(-b / 2a, f(-b / 2a))
Vertex of a quadratic
Axis of symmetry formula
-b / 2a
-b / 2a
Axis of symmetry formula
What happens if the x-intercept is not in the domain?
A hole in the graph at that x value
If f(x) = N(x) / D(x), and degree n < m, then
y = 0 is a horizontal asymptote
y = 0 is a horizontal asymptote of a function when
Degree n < m
If n = m, then
y = a / b. You divide leading coefficients and get the number
y = a / b. You divide leading coefficients and get the number
If n = m
If n > m, then
There is no horizontal asymptote
There is no horizontal asymptote when
n > m
How is a function odd or even?
Even when f(-x)=f(x)
Odd when f(-x)=-f(x)
How do you determine if a graph crosses a horizontal asymptote?
Set function equal to horizontal asymptote and solve
Set function equal to horizontal asymptote and solve
determine if a graph crosses a horizontal asymptote
How do you draw a number line of a rational function with vertical asymptotes?
Plot vertical asymptotes and input a number into x above and below horizontal asymptote . Greater than is plus and less than is minus
Plot vertical asymptotes and input a number into x above and below horizontal asymptote . Greater than is plus and less than is minus
draw a number line of a rational function with vertical asymptotes
For slant asymptotes, what do you do?
Divide the rational function. The remainder is put over the divisor, and f(x) is above and below the slant asymptote when this is greater than or less than 0.
Divide the rational function. The remainder is put over the divisor, and f(x) is above and below the slant asymptote when this is greater than or less than 0.
Slant asymptotes
How is f(x+h) - f(x) / h where
f(x) = 3-2x^2 written?
3-2(x+h)^2 - (3-2x^2) / h
What is the transformation that turns f into g?
f(x) = √x g(x) = √x+3
Shift left 3
Average rate of change formula
f(b)-f(a) / b-a
f(b)-f(a) / b-a
Average rate of change formula
How do you determine if a function touches or crosses the x-axis?
Odd multiplicity crosses, even touches
Wat determines end behavior?
Degree (even or odd)
Leading coefficient (pos. or neg.)
Direction of x (x—>+∞ or x—>-∞)
