Complex Numbers
Quadratic Functions
Analyzing Graphs
Zeros of Polynomials
Rational Functions
Inequalities
100
i2 = ?
-1
100
A quarterback throws a football to another player 40 yards away. The height of the ball, f(x), in feet, can be modeled by f(x) = -0.025x2 + x + 6, where x is the ball’s horizontal distance, in yards from the quarterback.
What is the ball’s maximum height and how far from the quarterback does this occur?
maximum (20,16). The maximum height is 16 feet. This height occurs after a horizontal distance of 20 yards.
100
This graph has ___ zeros.
4 unique zeros (x = -2, -1, 1, 3)
100
f(x) = 2x4 + 6x2 + 8. List all possible rational zeros
+-1, +-2, +-4, +-8, +-1/2
100
g(x)=(x^2 + 2x - 3)/(x - 3)
Domain:
{x|x!=3}
100
-x2 +5x - 6 < 0
(-oo, 2)uu(3,oo)
200
(3 - 8i) + (-2 - 4i)
1 - 12i
200
A quarterback throws a football to another player 40 yards away. The height of the ball, f(x), in feet, can be modeled by f(x) = -0.025x2 + x + 6, where x is the ball’s horizontal distance, in yards from the quarterback.
From what height did the quarterback throw the football?
y intercept (0,6). The quarterback threw the ball from a height of 6 feet.
200
This graph has the end behavior consistent with a polynomial of ____ (even/odd) degree and ____ (positive/negative) leading coefficient.
odd, positive
200
f(x) = 2x4 + 6x2 + 8
Use synthetic division to test a PRZ (possible rational zero). Based on your division, how do you know that this is not a root?
Not a root because the last number of the synthetic division is not equal to zero.
200
g(x)=(x^2 + 2x - 3)/(x - 3)
Why does this function not have a hole?
because the numerator factors to be (x + 3)(x - 1). There are no common factors in the numerator and denominator, so there can be no hole
200
(x+4)/((x+4)(x-7)) <0
{x|x<7 and x != -4}
OR
(-oo,-4)uu(-4,7)
300
5i(4 - 12i)
60 + 20i
300
A quarterback throws a football to another player 40 yards away. The height of the ball, f(x), in feet, can be modeled by f(x) = -0.025x2 + x + 6, where x is the ball’s horizontal distance, in yards from the quarterback.
If the football is not caught by someone, how far down the field will it go before hitting the ground? Round your answer to the nearest tenth of a yard.
x intercept (45.3,0). The ball will go 45.3 yards before hitting the ground.
300
This graph has ____ turning points. This means that the graph has a degree of at least ____.
6 turning points. at least 7th degree
300
f(x) = 2x4 + 6x2 + 8. Use Descartes’ Rule of Signs to explain why the function has no real roots
Positive Roots: no sign changes in f(x) so no positive real roots
Negative Roots: f(-x) = same as f(x) so no sign changes in f(-x) so no negative real roots
Therefore, all roots must be imaginary
300
g(x)=(x^2 + 2x - 3)/(x - 3)
Vertical Asymptote:
x = 3
300
solve the inequality
((x + 1)(x - 2))/(x - 1)>=0
[-1,1) uu [2,oo)
400
Simplify
4/(9-i)
(18+2i)/41
400
A quarterback throws a football to another player 40 yards away. The height of the ball, f(x), in feet, can be modeled by f(x) = -0.025x2 + x + 6, where x is the ball’s horizontal distance, in yards from the quarterback.
Sketch a graph of the function. Label the axes, x-intercept(s), y-intercept, and maximum point.
maximum (20,16)
x intercept (45.3,0)
y intercept (0,6)
x axis = horizontal distance (yds)
y axis = height of ball (ft)
400
Write a possible equation for the graph based on its end behavior, zeros (and their behavior), and turning points
(x+2)2(x+1)2(x-1)(x-3)2
400
f(x) = 2x4 + 6x2 + 8. Use your calculator to find all of the roots of the function f(x). Give exact answers where possible. Round other answers to two decimal places.
Rounded:
+-0.5 +-1.32i
Exact:
+-1/2 +-sqrt(7)/2i
400
g(x)=(x^2 + 2x - 3)/(x - 3)
Find the slant or horizontal asymptote. (I'm not telling you which one it is, you need to tell me and find it)
slant asymptote: y = x + 5
400
The function f(x) = 0.125x2 - 0.8x + 99 models a motorcycle’s stopping distance, in feet, traveling at x miles per hour on dry pavement. Determine the speeds that require a stopping distance of at least 267 feet.
40 mph or higher
500
if x2 + 8 = 0 then x = ?
+-2sqrt2i
500
For 9x2 - 7x + 12, f(x) --> ____ as x --> + infinity
f(x) --> infinity
500
f(x) --> ____ as x --> 4+
f(x) --> infinity
500
Write a polynomial of degree 5 that has no positive real zeros and only 1 negative real zero.
Simplest version example: x5 + 3 (must be all be same sign)
Complex version example: x5 + 3x4 + 9x2 + 7 (must be all same sign and only even degree terms or constant term after x5)
500
f(x) --> ____ as x --> - infinity
f(x) --> 1
500
(x^5 - 9x^2 + 15x - 2)/((x + 17)(x - 3))>=0
(-17,0.15]uu(3,oo)
**technically 0.146144... but rounded to two decimal places