Practice Test 79 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Let $z = 7 - 4i$. What is the complex conjugate $\overline{z}$? (Recall: the conjugate of $a+bi$ is $a-bi$.)
2. Order the following functions by eventual growth rate (from fastest to slowest) as $x\to\infty$: - Linear: $f(x)=20x$ - Quadratic: $g(x)=x^2$ - Polynomial (degree 5): $h(x)=x^5$ - Exponential: $p(x)=1.3^x$
3. A sequence can be viewed as a function whose domain is a subset of the integers. For the Fibonacci sequence defined on $n \in \{1,2,3,\dots\}$ by $f(1)=1$, $f(2)=1$, and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$, find $f(8)$.
4. Which of the following are factors of $p(x)=x^3-6x^2+11x-6$? Use the Factor Theorem (test $p(a)=0$).​
5. Prove the polynomial identity $a^2 - b^2 = (a + b)(a - b)$ by expanding the right-hand side and simplifying.
6. Rewrite the logarithmic equation $\log_5(125)=3$ in exponential form.​
7. What is the coefficient of $x^4 y^2$ in the expansion of $(x+y)^6$ (use Pascal's Triangle coefficients)?
8. Represent the system as a matrix equation $AX=B$ (variable order $x, y, z$): $\begin{cases} 2x+y-z=5\\ x-3y+2z=1\\ 4x+y+z=9 \end{cases}$​
9. Use polynomial long division to express $\frac{x^4 + 1}{x - 1}$ in the form $q(x) + \frac{r(x)}{x-1}$, where $\deg(r) < 1$.
10. Let the parent function be $f(x)=x^2$. Which function represents a shift of the graph of $f$ right 3 units and down 2 units?
11. Let $z=2-5i$. What is $\overline{\overline{z}}$? (Recall that taking the conjugate twice returns the original complex number.)
12. A streaming channel has 2,500 subscribers and grows by 12% each month. Model the situation with an exponential equation and solve for the number of months $t$ until the channel reaches 5,000 subscribers. Let $t$ = number of months.
13. Use technology to find $A^{-1}$ and then compute $X=A^{-1}B$. Given $A=\begin{bmatrix}1&2&0\\0&1&1\\2&0&1\end{bmatrix},\quad X=\begin{bmatrix}x\\y\\z\end{bmatrix},\quad B=\begin{bmatrix}5\\1\\4\end{bmatrix}.$ Solve $AX=B$ using the inverse matrix method (do not hand-calculate the $3\times 3$ inverse).
14. A small business models its daily profit (in dollars) as a function of the number of items sold, $p(x)=-(x-5)^2+16$, where $x$ is items sold. On what interval is $p(x)$ increasing?
15. Subtract and express in standard form $a+bi$: $(6-4i)-(1+9i)$.
16. Solve the linear-quadratic system algebraically (by substitution). The solutions correspond to the intersection point(s) of the line and the parabola. $\begin{cases} y = x + 1 \\ y = x^2 - 2x + 1 \end{cases}$ What are all solutions $(x,y)$ to the system?​
17. A savings account is modeled by the function $A(t)=2500(1.04)^t$, where $t$ is in years. What is the constant percent rate of change, and is it growth or decay?
18. A circle has center at $(3, -2)$ and passes through the point $(7, 1)$. If this circle is reflected across the line $y = x$, what is the equation of the reflected circle?
19. A cyclist rides 18 miles at a speed of $r$ mph and then 12 miles at a speed of $(r-3)$ mph. The total travel time is 1.5 hours. Set up and solve a rational equation to find the cyclist's initial speed $r$. Let $r$ = the speed (in mph) for the first part of the trip.
20. Use the identity $(a+b)^2=a^2+2ab+b^2$ to compute $101^2$ mentally by rewriting $101$ as $100+1$.

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