real zeroes
State the zeroes of the following quadratic
f(x)=(x-1)(x+3)
Zeroes occur at x=1 and x=-3
State the constant of variation and the power for the below power function
F=-kx
power is 1, constant of variation is -k
Find x-int, y-int, & asymptotes for the following function
f(x)=1/(x-2)
no x-int, y-int at (0,-0.5)
vertical asymptote @ x=2, horizontal @ y=0
State the zeros of the following polynomial:
f(x)=(x-i)(x+i)(x-4)
Zeros occur at x=i, x=-i, and x=4
Find the vertex and axis of the graph of the quadratic f(x)=1/3(x+2)^2-3
Vertex: (-2,-3)
Axis: x=-2
Find all zeroes of the following quadratic
f(x)=x^2+2x+1
The only zero occurs at
x=-1
State the power & constant of variation for the function:
f(x)=1.7x^(2/3)
power: 2/3
constant of variation: 1.7
Describe how to transform the reciprocal function f(x)=1/x into the function g(x)=-1/(x+2)
reflect f(x) across the x-axis and translate 2 units to the left to obtain g(x)
Find all zeros of the following quadratic:
f(x)=x^2+2x+4
x=-1\pm sqrt(3) \ i
Write the following quadratic equation in vertex form f(x)=2x^2+4x+8
f(x)=2(x+1)^2+6
Is (x-4) a factor of the following function
f(x)=x^4-6x^3+4x^2+30x-45
No! f(4)=11 so (x-4) is not a factor!
What is the constant of proportion & exponent for the following power function:
f(x)=2/3root(5)(x^2)
constant of proportion: 2/3
exponent: 2/5
Let f(x)=-2/(x^2+3x) . Which of the following values have to be excluded from the domain of f ?
a) only 0
b) only 3
c) only -3
d) only 0, 3
e) only 0, -3
Write a degree 3 polynomial in factored form which has zeros 2+2i and 3.
(x-(2+2i))(x-(2-2i))(x-3)
Write the equation of a parabola which has a vertex at (2,3) and contains the point (-1,-4)
f(x)=-7/9(x-2)^2+3
State the degree of the polynomial, find all zeroes, state their multiplicities, and say whether or not the function crosses the x-axis at each zero
f(x)=(x-1)(x+2)^2(x-3)^3
f(x) is degree 6
zeroes: x=1, multiplicity 1, f(x) crosses at x=1
x=-2, multiplicity 2, f(x) does not cross at x=-2
x=3, multiplicity 3, f(x) crosses at x=3
Write the statement as a power function equation, using k as the constant of variation:
The volume V of a circular cylinder with fixed height is proportional to the square of its radius r.
V(r)=kr^2
Find x-int, y-int, vertical asymptote, & end behavior asymptote for the following function
f(x)=(x^2-3x-7)/(x-3)
end behavior asymptote q(x)=x, vertical asymptote at x=3
y-int at (0,7/3), x-intercepts occur at
x=(3\pm\sqrt37)/2
Write a degree 3 polynomial in expanded form which has zeros 3-i and 5.
x^3-11x^2+40x-50
Find intercepts & asymptotes, use limits to describe the behavior at the vertical asymptotes, & state the domain & range of the following function:
f(x)=(x^2+2x-3)/(x+2)
Intercepts: (-3,0), (1,0), (0,-3/2)
asymptotes: x=-2, y=x
domain: x≠2, range (-∞,∞)
lim_(x\to-2^-)f(x)=\infty \, lim_(x\to-2^+)f(x)=-\infty
Using the Rational Zeros theorem & your knowledge of factoring polynomials, list all possible rational zeros then completely factor f(x):
f(x)=4x^4-7x^3-17x^2+35x-15
possible zeros: (\pm1,\pm3,\pm5,\pm15)/(\pm1,\pm2,\pm4)
f(x)=(x-1)(4x-3)(x+sqrt5)(x-sqrt5)
Write a sentence that expresses the relationship in the formula, using the language of variation or proportion.
d=p^2/(2g) ,
where d is the distance traveled by a free-falling object dropped from rest, p is the speed of the object, and g is the acceleration due to gravity.
The distance traveled by a free-falling object dropped from rest is proportional to the square of the object's speed.
Find the intercepts, vertical asymptotes, & end behavior asymptote:
f(x)=(2x^5+x^2-x+1)/(x^2-1)
x-int at (-1.108,0), y-int at (0,-1)
vertical asymptotes at x=-1, x=1
end behavior asymptote:
q(x)=2x^3+2x+1
Write the function as a product of linear & irreducible quadratic factors all with real coefficients
f(x)=x^4+3x^3-3x^2+3x-4
f(x)=(x-1)(x+4)(x^2+1)
Let f(x)=x^4+2x^3-11x^2-13x+38
Use the upper & lower bound tests to prove that all of the real zeros of f lie on the interval [-5,4].
Find all of the rational zeros of f & factor f(x) using those zeros.
Last line of synthetic div for f(x)/(x-4) all non-negative, last line of synthetic div for f(x)/(x+5) alternate btw non-neg & non-pos
Only rational zero occurs at x=2 -->
f(x)=(x-2)(x^3+4x^2-3x-19)