Chapter 2 (2.1 to 2.6) Flashcards
What is the standard form of a quadratic and how would you determine its axis of symmetry, y/x intercept(s), vertex, max/min value, and appearance?
f(x) = ax^2 + bx + c vertex is (-b/2a) for x, plug in for y (axis is x=x-corr) Y intercept is c, or plug in 0 for x Plug in 0 for y to find x intercepts if a>0 it opens up + min if a 1 skinny |a|
What is the vertex form of a quadratic and how would you determine its axis of symmetry, y/x intercept(s), vertex, max/min value, and appearance?
f(x) = a(x - h)^2 + k Vertex is (h, k) Plug in 0 for x to find y intercepts Plug in 0 for y to find x intercepts if a>0 it opens up + min if a 1 skinny |a|
What would a function of the first degree look like as an equation, in a graph, and written down?
Y=ax+b
Line starting down, ending up (a>0), starting up, ending down (a
What would a function of the second degree look like as an equation, in a graph, and written down?
Y=ax^2+bx+c
Single curve starting up, ending up (a>0), starting down, ending down (a
What would a function of the third degree look like as an equation, in a graph, and written down?
Y=ax^3+bx^2+cx+d
Double curve starting down, ending up (a>0), starting up, ending down (a
What would a function of the fourth degree look like as an equation, in a graph, and written down?
Y=ax^4+bx^3+cx^2+dx+e
Triple curve starting up, ending up (a>0), starting down, ending down (a
What would a function of the fifth degree look like as an equation, in a graph, and written down?
Y=ax^5+bx^4+cx^3+dx^2+ex+f
Four curves starting down, ending up (a>0), starting up, ending down (a
How would you divide (x^3-2x^2-9) by (x-3)?
Multiply x-3 to cancel with the first number and add, if it is a factor you will end with 0, if not put the remainder over (x-3)
x^2+x+3
x-3/x^3-2x^2+0x-9
x^3-3x^2
x^2+0x
x^2-3x
3x-9
3x-9
0
How would you synthetically divide (x^3-2x^2-9) by (x-3)?
3] 1 -2 0 -9
3 3 9
1 1 3 0
1x^2=+1x+3
How would you synthetically substitute 2 into 2x^6 + 3x^4 - x^2 +3?
2] 2 0 3 0 -1 0 3
4 8 22 44 86 172
2 4 11 22 43 86 175
What is the square root of -1, -2, and -12?
i, 2i, 2i times the square root of 3
What is i^2?
-1
What are complex numbers made of?
real and imaginary numbers (ex: a +bi. a is real and b is imaginary
What is i to the 1st, 2nd, 3rd, and 4th power?
i, -1, -i, 1
How would you find i to the 103rd power?
find the remainder of 103/4 and count up (ex: remainder is 3 so it equals i to the 3rd or -i)
What is the rational zero test?
every rational zero of a polynomial will have the form p/q where p is a factor of the constant and q is a factor of the leading coefficient
What are the rational zeros of 2x^3 + 3x^2 - 8x +3?
Ps are +/- 1 and +/- 3
Qs are +/-1 and +/- 2
P/Qs are +/- 1 and +/- 3 and +/- 1/2 and +/- 3/2
Write a function with the zeros of 4 and -3i
(x - 4)(x + 3i)(x - 3i)
If something is a polynomial with an irrational zero, it must have a pair of conjugates so they can cancel
What is Descartes Rule of Signs?
The number of positive real zeros of a function is either equal to the number of variations in sign of f(x) or less than that number by an even integer
The number of negative real zeros of a function is either equal to that number of variations in sign of f(-x) of less than that number by an even integer
What are the possible numbers of positive and negative real zeros of -2x^3 + 5x^2 - x + 8?
F(x) is 3 or 1 positive real zeros (- + - +)
F(-x) is 0 negative real zeros (+ + + +)
What are the upper and lower bound rules?
When f(x) is divided by (x - c)
If c > 0 and each # is the last row is + it is an upper bound
If c
What is a rational function and how would you find vertical and horizontal asymptotes?
VA- set the denominator equal to zero and solve
HA- (n degree of numerator and m degree of denominator) if n>m none, n=m leading coe/leading coe, if n
How would you find the zeros or x-int of a rational function?
Set numerator equal to zero and solve, watch domain
