Polynomial & Rational Functions
What is the maximum number of extrema (local minima and maxima) that a polynomial function of degree $n$ can have?
n-1
If the population of a bacteria culture triples every 4 hours, write an exponential expression representing the population after $t$ hours, given an initial population P0
P(t) = P0x3^t/4
State the amplitude, period, and midline of the function:
Amplitude = 3, Period = pi, Midline: y = 5
Convert the polar coordinates (4, 5pi/6) into rectangular coordinates (x, y).
(-2rad3, 2)
Find the 20th term of the arithmetic sequence defined by:
62
Find the equation of the vertical asymptote and the coordinate of the hole for the rational function:
f(x)=(x-3)(x+2)/(x-3)(x+5)
Solve for x:
x=3
Evaluate exactly:
sin (arccos(-4/5))
3/5
Eliminate the parameter t to find the rectangular equation for the parametric equations
x(t) = 3t - 1 and y(t) = t^2 + 2
y=(x+1/3)^2 or y= 1/6x^2 + 2/9x + 19/9
Find the exact sum of the infinite geometric series:
16
If a polynomial function $P(x)$ has a leading term of $-4x^5$, describe its end behavior using limit notation.
lim_{x \to \infty} P(x) = -infty and lim_{x \to -\infty} P(x) = infty
In a semi-log plot where the vertical axis is logarithmic (base 10) and the horizontal axis is linear, a data set forms a perfect straight line with a slope of 2 and a vertical intercept of 3. What type of function models the original data?
An exponential function
Solve the equation 2\sin^2(x) - sin(x) - 1 = 0 for 0<x<2pi
x=pi/2, 7pi/6, 11pi/6
An object moves in the plane with a constant velocity vector v=(-3,4). If the object starts at the point (2, 5) at time t = 0, write the parametric equations for the object's position at time t.
x(t) = 2 - 3t and y(t) = 5 + 4t
For a function f(x), if the average rate of change over the interval [2, 5] is 4, and the average rate of change over the interval [5, 9] is 4, what can you conclude about the rate of change of f(x), and what type of function is it?
The rate of change is constant, meaning f(x) is a linear function.
A rational function R(x) has a numerator of degree 3 and a denominator of degree 2. What type of end behavior model does this function have?
It has a slant (oblique) asymptote
Let f(x) = 5xe^2x. Find an expression for its inverse function, f^{-1}(x).
f^{-1}(x) = 1/2 ln(x/5)
A Ferris wheel has a diameter of 40 meters and turns at a constant speed, completing one full rotation every 6 minutes. Passengers board at the lowest point, which is 2 meters above the ground. Write a sinusoidal function h(t) for a rider's height above the ground after t minutes.
h(t) = -20 cos(pi/3 t) + 22
Convert the rectangular equation x^2 + y^2 - 6y = 0 into a polar equation isolated for r. What shape does this graph form?
r=6sin(0), it forms a circle
If the total revenue of a company is modeled by a quadratic function, describe how the average rate of change of the revenue changes over equal-length consecutive intervals.
The average rates of change will form an arithmetic sequence (they change at a constant rate).
Find a possible cubic polynomial formula for $f(x)$ that has zeros at x = -2 (multiplicity 2) and x = 3 (multiplicity 1), given that the y-intercept is (0, -24)
f(x) = 2(x+2)^2(x-3)
A beverage is cooling in a room that is 20C. According to Newton’s Law of Cooling, the difference in temperature decreases exponentially. If the beverage cools from 90C to 55C in 10 minutes, write the function T(t for the temperature of the drink after t minutes.
T(t) = 20 + 70(1/2)^t/10 or T(t) = 20 + 70e^{-0.0693t}
Simplify the expression sec²(0)-1/sec²(0) into a single trigonometric function.
sin^2(0)
Describe the behavior of the polar curve r = 3 - 4 sin (0) when 0=pi/2. Is the curve moving closer to or farther from the pole at this instant?
At 0=pi/2, r=-1
Write a recursive definition for a geometric sequence where the third term a3 = 18 and the sixth term a6 = 486.
a1=2 and aN=3xaN-1