Math 112 Exam 3

Quadratic functions

100

find the vertex of 

f(x)= x²-6x+5

(3, -4)

200

Find the maximum or minimum value of 

f(x)= -x²+4x + 1

maximum value 

5

300

Find the x intercepts of f(x)= x²-9

(-3,0), (3,0)

400

write in vertex form

f(x)= x² +8x + 3

f(x)= (x +4)² -13

500

a ball's height is defined by the function 

h(t)= 40t- 16t²

What is the balls maximum height?

25 ft

Polynomial Graphs

100

f(x)= x³

Describe the end behavior: 

as x--> -infinity, y-->______

as x--> +infinity, y--> ______

-infinity,

+infinity

200

What is the end behavior of 

f(x)= x(1-x)(x+5)³

as x--> -inf, y--> +infinity

as s--> +infinity, y--> -infinity

300

Find the zeroes of

f(x)= (x-3)(x+1)(x+5)

What do they do at each? (bounce or go throught)

x= 3,-1,-5

all go through; odd exponents

400

Determine the multiplicity of the zero in 

f(x)= (x-2)⁴

What will the x do at this zero (bounce or go through)

multiplicity=4 (x=2)

Bounce

500

Which graph is represented by

y=(x-2)²(x+4)

red; a squared zero will have a "sharper" end behavior than a zero to the 6th power

Factoring and zeroes

100

Factor f(x)= x²-16

how does this help us with determining the graphs of polynomials?

f(x)=(x+4)(x-4)

we can see what the zeros of the graph will be more clearly in this form

200

Find the zeros of 

F(x)= x²-5x +6

x=2, x=-3

300

Factor the polynomial and use the factored form to find the zeros

f(x)= x⁵-4x³

What do the zeroes do at the x-axis? which zero will have a flatter line than the others?

f(x)=x³(x-2)(x+2)

x=0,-2,2

all cross through; x=0 will be flatter because it is cubed

400

Factor

f(x)= x³-4x² -4x+16

f(x)=(x-2)(x+2)(x-4)

500

Factor and list all zeros

f(x)= x³+3x²-6x +18

f(x)=(x²-6)(x+3)

x= +square root(6), -square root(6), x=-3

Polynomial Division

100

Divide 

(x²+5x+6)/x+2

Use either synthetic, OR factor the top into (x+3)(x+2) to cancel x+2 on top and bottom; 

x+3

200

divide and express in the form 

P(x)= D(x) x Q(x) + R(x)

P(x)= x³ +8x²-2x + 4, D(x)=x-2

P(x)=__________

P(x)= (x-2)(x² +10x +18) + 40

300

Divide 

P(x)= x⁴ -x³ + 5x +1 

D(x)= x²+4

what is q(x), what is r(x)

x²-x-4, R(x)= 9x +17

400

Divide

9x³ + 3x² +26x

3x² +2

quotient= 3x +1 

remainder= 20x-2

500

Divide 

x² +3x

x-4

x +7, remainder 28

Rational Functions

100

Does this graph have a horizontal asymptote, and what is the vertical asymptote

f(x)= 1/(x-3)

yes; y=0

v.a at x=3

200

What is the horizontal asymptote

f(x)= 2x+1/x

what is the vertical asymptote

h.a at y=2

v.a at x=0

300

Find the slant asymptote if there is one.

f(x) = x²+3x/ x-4

what kind of function is the slant

linear

y=x+7

400

Find the horizontal and vertical asymptotes, x and y intercepts, and holes

r(x)= (x² -4x +3)/(x²-5x+6)

hole: x=3

x intercept (1,0)

y intercept (0,1/2)

v.a: x=2

h.a: y=1

500

find horizontal and vertical asymptotes (if any), find x and y intercepts, and holes

r(x)= (x²-9)/(x²-4x +3)

hole: x=3

x intercept: (-3,0)

y intercept: (0, -3)

v.a.: x=1

h.a. y=1