Unit 1: Polynomial & Rational Functions
A polynomial satisfies
f(2)=0
f(3)=0
f(4)=0
Write the expanded form of the polynomial
x3-9x2+26x-24
Given the function:
f(t)=1000 (1.005)t
Find the percent growth rate per year.
0.5%
Without using a calculator, determine the value of
sin (cos-1(-√2/2))
√2/2
Let f(x)=x2+4x+1.
Find f(i).
f(i)= 4i
Find the 5th term of the geometric sequence:
a1=3, r=−2
=-48
(Formula: an=3(-2)4=3(16)=48
But since the pattern alternates signs:
3, -6, 12, -24, 48)
Consider the rational function
f(x)= 2x3 + 5x / x2 - 1
What type of asymptote does the function have?
A slant asymptote
Find the inverse of
f(x)= ex-3
f-1(x)=ln(x) + 3
State the domain and range of the following function: f(x)=arccos(x)
Domain: [-1,1] Range: [0, pi]
Given the polynomial p(x) = (x-3)2(x+5), what is the multiplicity of the zero at x=3, and what effect does that multiplicity have on the graph?
Multiplicity of 2; the graph bounces on x = 3.
Find the value of x:
4x = 16√2
9/4
A polynomial has degree 5 and a positive leading coefficient.
Describe the end behavior.
As x→−∞, f(x)→−∞
As x→∞, f(x)→∞
(Down on the left, up on the right.)
Solve:
log5(x)+log5(4)=2
x = 25/4
Solve for x for the following equation: cos(x)-sin(x)=0
x=pi/4 (45degrees)
The function g(x) is a periodic function with a period of 6. If the domain of g includes all real numbers, what is the value of: g(27) - g(3)
0
(Since the period is 6, the function repeats its output every 6 units (g(x) = g(x + 6n)). Because 27 = 3 + (4 * 6), it follows that g(27) = g(3). Therefore, g(27) - g(3) = 0.)
abs (x-3) < 5
Give your answer in interval form.
A polynomial function p is given by
p(x) = −x (x - 4)(x + 2).
What are all intervals on which p(x) ≥ 0?
(− ∞, -2] ∪ [0, 4]
A colony of bacteria decays so that the population t days from now is given by A(t) = 1000 (1/2)1/4
a. What is the original amount?
b. How much will be present in 4 days?
c. What is the half-life?
a. 1000
b. 500
c. 4 days
In triangle ABC, side a = 7, side c = 10, and angle B = 60degrees. Find the length of side b.
b = square root of 79 (approx 8.99)
Simplify the following expression: {sin(x)*sec2(x)}/csc(x)
tan2(x)
A ball is attached to a 5-meter string and moves in a circle. Its height above the ground can be modeled by:
h(t) = 4 + 5sin(t)
Find the maximum height and minimum height of the ball.
Maximum height: 9 meters
Minimum height: -1 meters
The function f(x)=2x+5/x−3
Find the equation of any vertical asymptote, horizontal asymptote, and the x-intercept and the y-intercept of the function.
Vertical asymptote: x=3
Horizontal asymptote: y=2
x-intercept: x=−5/2
y-intercept: y= -5/3
Solve for x:
log3(x−2)+log3(x−4)=2
x=5
In triangle ABC, angle A = 30degrees, side a = 10, and side b = 16. How many distinct triangles can be formed?
Two distinct triangles.
(This is the Ambiguous Case (SSA). The height of the triangle is $h = 16 \sin(30^\circ) = 8. Since the opposite side (a=10) is greater than the height (8) but less than the adjacent side (b=16), two different triangles can be drawn.)
Solve for x on 0≤x<2pi:
2cos2(x)−cos(x)−1=0
x=0, 2pi/3, 4pi/3
1/2(logbM + logbN - logbP)
logb√MN/P