Linear and Quadratic Functions
Rewrite the function in slope intercept form:
6=2x+y
y=-2x+6
Polynomials with an odd leading coefficient and an odd degree have the end behavior:
up on the left and down on the right
Factor R(x).
R(x)=frac{x^2-9}{x^2-4x-5}
frac{(x-3)(x+3)}{(x-5)(x+1)}
Zeros, __________, ___________, and ____________ all mean the same thing.
Roots, x-intercepts, solutions
Add these complex numbers:
(2-5i)+(2i+2)
-3i+4
The equation of a linear function is given. State the slope and y-intercept.
y=frac{2}{3}x-7
m=frac{2}{3}, b=-7
Polynomials with an even leading coefficient and an even degree have end behavior:
up on the left and up on the right
Identify any restrictions on the domain for R(x). Give your solution in interval notation.
R(x)=frac{x^2-9}{x^2-4x-5}
(-oo, -1)U(-1,5)U(5, oo)
Complete the statement: if f(c)=0, then ______ is a factor of f(x).
(x-c)
Multiply (-2-i) with its conjugate.
5
Given the function stated below, find the average rate of change on the interval [1,4] and then give the interval where the function is increasing and the interval when it is decreasing.
6=2x+y
average rate of change is the slope, m=-2
since the slope is negative, the function is never increasing and always decreasing. The interval for decreasing is:
(-oo, oo)
Consider the function f(x). What are the zeros of f(x)?
f(x)=x(x-2)^5(3x-1)^2
x=0, 2, frac{1}{3}
State the vertical asymptote(s) and any hole(s) in R(x).
R(x)=frac{x^2-9}{x^2-4x-5}
No holes, VA:
x=5, x=-1
Using Rational Root Theorem (RRT), list all possible roots for f(x).
f(x)=2x^3-3x^2-8x+12
+-1,2,3,4,6,12,frac{3}{2}
Suppose a polynomial has a zero, 3+i. Write the complex conjugate of this zero.
3-i
Use f(x) provided to state the axis of symmetry, find the intervals of increase and decrease, state the vertex and identify the vertex as max/min.
f(x)=-(x-1)^2+2
AOS: x=1, increasing: (-infinity, 1), decreasing: (1, infinity), vertex: (1,2) and it is a max
Consider the function f(x). Determine the maximum turning points of f(x) and give the end behavior.
f(x)=x(x-2)^5(3x-1)^2
TP: 8-1=7
up and up
State the x-intercept(s), y-intercept and horizontal asymptote for R(x). Give your solutions for intercepts as ordered pairs and your HA as the equation of a line.
R(x)=frac{x^2-9}{x^2-4x-5}
(3,0), (-3,0),(0, frac{9}{5})
y=1
Use Descartes Rule of Signs to give the number of positive and negative real zeros for f(x).
f(x)=2x^3-3x^2-8x+12
pos: 2 or 0
neg: 1
Suppose a polynomial function has zeros: 2, -1, 3+i. Write this 4th degree polynomial function in factored form.
f(x)=(x-2)(x+1)(x-(3+i))(x-(3-i))
Use f(x) provided to state the axis of symmetry, find the intervals of increase and decrease, state the vertex and identify the vertex as max/min.
f(x)=x^2+6x+5
AOS: x=-3, increasing: (-3, infinity), decreasing: (-infinity, -3), vertex: (-3,-4) and it is a minimum
Consider the function f(x). What are the multiplicities of zeros and what is the graph effect?
f(x)=x(x-2)^5(3x-1)^2
x=0, multiplicity 1, cross
x=2, multiplicity 5, cross
x=1/3, multiplicity 2, bounce
Solve the inequality:
frac{(x-3)(x+1)}{x-2}>0
(0,2)U(3,oo)
Use synthetic division to fully factor f(x). Then state the zeros of f(x).
f(x)=2x^3-3x^2-8x+12
f(x)=(x-2)(x+2)(2x-3)
x=2,-2,frac{3}{2}
Fully expand the factored form of f(x) and write it in standard form. (multiply it out)
f(x)=(x-2)(x+1)(x-(3+i))(x-(3-i))
a(x^4-7x^3+14x^2+2x-20)