Polynomial Functions

polynomial functions and end behavior

100

sketch a 4th degree polynomial with a negative lead coefficient

should look like an 'M'

200

evaluate p(x) = 3x + x^2 for p(-2b)

-6b + 4b^2

300

describe the end behavior of a 3rd degree polynomial with a positive lead coefficient

as x approaches infinity, f(x) approaches infinity as x approaches negative infinity, f(x) approaches negative infinity

400

graph f(x) = x^4 - 7x - 3, estimate the x-coordinates for the relative max and min.

minimum about 1

500

graph y = 2(sq rt(x)) and identify the domain and range.

see graph, domain: x > 0, y > 0

zeros

100

what are the maximum and minimum number of real zeros that a cubic polynomial function could have?

3 and 1

200

sketch a quartic function with a negative lead coefficient and 2 real zeros

'M' shape, only crossing x-axis twice

Operations

100

f(x) = 3x + 1 and g(x) = 2x - 3 Find (f - g)(x) and (fg)(x)

(f-g)(x) = x + 4 (fg)(x) = 6x^2 - 7x - 3

200

f(x) = x^2 + 5 and g(x) = 3x Find (f/g)(x) and (f + g)(x)

(f/g)(x) = (x^2 + 5)/ 3x where x is not = 0 (f + g)(x) = x^2 + 3x + 5

300

g(x) = 2x and h(x) = x + 2 Find g[h(x)]

g[h(x)] = 2x + 4

400

g(x) = x + 4 and h(x) = x^2 - 1 Find g(h(-3))

g(h(-3)) = 12

500

f(x) = 3x and g(x) = x + 4 and h(x) = x^2 - 1 Find f(g(h(x)))

f(g(h(x))) = 3x^2 + 9

inverse functions

100

Find the inverse function of f(x) = 2x - 4

f^-1(x) = (x + 4)/2

200

Verify that f(x) = 2x - 3 and g(x) = .5x + 1.5 are inverses.

show graphs are reflected in y = x line...or... show f(g(x)) = g(f(x))

300

Graph y >sq rt(2x + 4)

x = -2, -1, 0, 1 Shade above dotted line y = 0, 1.4, 2, 2.4

400

how is the graph of y = sq rt(x - 3) + 1 related to y = sq rt (x) ?

it is shifted to the right 3 units and up 1.

500

Find the inverse of f(x) = 3x^2 - 7

f^-1(x) = sq rt((x + 7)/3), x > 7 or [7, inf)

solving for zeros

100

put in quadratic form: 24x^8 + x^4 + 4

24(x^4)^2 + (x^4) + 4

200

solve: x^4 - 7x^2 + 12 = 0

2, -2, sq rt(3), - sq rt(3)

300

find all the zeros of x^3 - 6x^2 + 13x - 10

2, 2+i, 2-i

400

Find all the zeros of x^4 + 5x^3 + 9x^2 + 45x

-5, 0, 3i, -3i