Graph-ically Speaking
what is the end behavior?
f(x)=x2
right/left ends go up
F(x) = x3
F(x) = (x-6)3
shifted right 6 units
Is it a polynomial?
๐น(๐ฅ) = ๐ฅ2 โ 4 / ๐ฅ5 + 3
No (negative exponent)
(m2 โ 7m โ 11) รท (m โ 8)
m + 1 - 3/(m-8)
-6
what is the end behavior?
f(x) = -4x22 + 6
falls to left/right
F(x) = (x)
F(x) = (-2x) โ 5
stretched horizontally by a factor of 1/2, reflected over the y-axis, and shifted down 5 units
What are the degree and leading coefficient?
๐(๐ฅ) = 3 โ 5x6 + 2x
Degree: 6
Leading Coefficient: -5
(x2 + 5x + 3) รท (x + 6)
x โ 1 + 9/(x+6)
g(a) = 3a + 2
f(a) = 2a โ 4
Find (g /f) (3)
11/2
What is the end behavior?
F(x) = x3 - 2x2 + 7
right rises, left falls
f(x) = x3
g(x) = 3(x + 1)3
expand vertically by a factor of 3, shift left 1 unit
What is the end behavior?
f(x) = -3(x-4)2(x+7)3
Falls to the right; rises to the left
x3 โ 13x2 + 40x + 18 รท (x โ 7)
x2 โ 6x โ 2 + 4/(x โ 7)
-7
Where does the graph touch/cross?
f(x) = x2(3x-4)(x+5)4(2x-8)3
x=0, touches
x=4/3, crosses
x=-5, touches
x=4, crosses
f(x) = x2
Transformations:
expand vertically by a factor of 3, shift down 3 units
f(x) = 3(x)2 - 3
Solve by factoring:
๐ฅ3 + 3๐ฅ2 โ 5๐ฅ โ 15 = 0
(x2 - 5)(x + 3) = 0,
x = +/- sqrt 5, -3
(6v3 - 56v2 + 106v - 57)/(6v -8)
v2 - 8v + 7 โ 1/6v - 8
g(n) = 3n + 2
f (n) = 2n2 + 5
Find g(f(2))
41
what does the graph look like?
f(x) = x2(x-4)3(2x+8)5
touches @0, crosses @4, crosses @-4
f (x) = [ x ]
expand horizontally by a factor of 2, shift right 1 unit, shift up 3 units
[1/2 (x - 1)] + 3
What are the zeroes and their multiplicity?
g(x) = 8x3(x+3)2(2x-5)(3x-7)4
Zeros:
x = 0, mult. 3
x = -3, mult. 2
x = 5/2, mult. 1
x = 7/3, mult. 4
(3x4 - 6x3 - 27x2 + 33x + 61) / (3x - 9)
x3 + x2 - 6x - 7 โ 2/3x - 9
f (n) = 2n
g(n) = โn โ 4
Find ( f o g)(n)
-2n - 8