Expert Answers to Algebra 2 Problems

How do exponential functions differ from polynomial and rational functions?

The following question consider the Gompertz equation, a modification for logistic growth, which is often used for modeling cancer growth, specifically the number of tumor cells. [T] The Gompertz equation has been used to model tumor growth in the human body. Starting from one tumor cell on day 1 and assuming α=0.1α=0.1 and a carrying capacity of 10 million cells, how long does it take to reach “detection” stage at 5 million cells?

Rewrite as an exponential equation
${\displaystyle \ln 5=y}$

Determine the slope of the line represented by the linear equation 5x+6y=40
1.5
2.6
3.${\displaystyle \frac{5}{6}}$
4.${\displaystyle \frac{-5}{6}}$

Graph the rational functions. $y=\frac{x}{(x^2)-2}$

Determine which type of function-linear, quadratic, or exponential-would be best for modeling each situation below. You do not need to write an equation for the situation. a. Bailey is growing bacteria on a petri dish for science class. The population of bacteria doubles every 20 minutes, so the initial number of bacteria is 1000, then 2000, then 4000, then 8000, etc. b. Avery is working on her math homework. She finished nine problems in class. She knows that, on average, she can do three problems in five minutes. c. Keiton loves catapults. He is building a small catapult that will launch an apple into the air and over the fence into his friend 's backyard.

For each polynomial, state the constant term, the leading coefficient, and the degree of the polynomial.
$5x^2-8x+2x^4-3$

In the following equation/s, find the indefinite integral :-
$\int \frac{x^2}{x-1}dx$

Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: $P(x)=a(x-\alpha )(x-\beta )(x-\gamma ),a\ne 0$
Type 2: Two real zeros, one repeated: $P(x)=a(x-\alpha )2(x-\beta ),a\ne 0$
Type 3: One real zero repeated three times: $P(x)=a(x-\alpha )3,a\ne 0$

Type 4: One real and two imaginary zeros: $P(x)=(x-\alpha )(ax2+bx+c),\mathrm{△}=b2-4ac<0,a\ne 0$
Experiment with the graphs of Type 4 cubics. What is the geometrical significance of $\alpha \alpha$ and the quadratic factor which has imaginary zeros?

(a) How do you find vertical asymptotes of rational functions? (b) Let s be the rational function ${\displaystyle s\left(x\right)=\frac{a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0}{b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0}\mathrm{\$}\mathrm{\$}}$
How do you find the horizontal asymptote of s? (c) Find the vertical and horizontal asymptotes of ${\displaystyle f\left(x\right)=\frac{5x^2+3}{x^2-4}}$

You decide to start saving pennies according to the following pattern. You save 1 penny the first day, 2 pennies the second day, 4 the third day, 8 the fourth day, and so on. Each day you save twice the number of pennies you saved on the previous day. Write an exponential function that models this problem. How many pennies do you save on the thirtieth day?

For the following exercise, for each polynomial, a. find the degree; b. find the zeros, if any; c. find the y-intercept(s), if any; d. use the leading coefficient to determine the graph’s end behavior; and e. determine algebraically whether the polynomial is even, odd, or neither.
${\displaystyle f\left(x\right)=-3x^2+6x}$

Which of the options is the differential equation that accepts the solution of the following equation?
$y(x)=c_1\cos x+c_2\sin x$
y+y=0
y+y=0

Factor each polynomial. If a polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary.:
$7a^3+14a^2$

Factor completely.
(c)$9x^2-48x+64$
(d)$4x^2+1$

Solve the system $Ax=b$ using the given LU factorization of A
$A=\left[\begin{array}{cc}-2& 1\\ 2& 5\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]\left[\begin{array}{cc}-2& 1\\ 0& 6\end{array}\right],b=\left[\begin{array}{c}5\\ 1\end{array}\right]$

Solve the given equation using linear differential equations.
$y^{\prime }=\csc x+y\cot x$

Solve for x and y.
3x+12y=-30...(1)
6x+6y=-24...(2)

Consider the following system of linear equations:
4x+2y=25
4x-y=-5
If the value of y is 10 what is the value of X for this system:
1.1.25
2.11.25
3.1.45
4.5

To find: The angle of congruence.