Practice Test 19 - Algebra 2

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

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1. On the complex plane, the distance between two complex numbers $z$ and $w$ is $|z-w|$. What is the distance between $z=3+4i$ and $w=1+i$?
2. A function is sampled at integer inputs, producing the sequence $2, 6, 18, 54, \dots$. Which statement best classifies the growth and connects it to a function type?
3. A ball follows the path $y=-x^2+6x$ (a parabola). A laser beam follows the line $y=2x$. Where do they intersect? Solve the system. $\begin{cases} y = -x^2 + 6x \\ y = 2x \end{cases}$
4. Let the parent function be $f(x)=x^3$. What is the effect on the graph of replacing $f(x)$ with $g(x)=f(x+2)$?
5. Vector addition of complex numbers corresponds to adding their coordinates on the complex plane. Using $a+bi\leftrightarrow (a,b)$, what is the sum $(2+i)+(1+3i)$ and which point represents it?
6. A culture of bacteria starts with 300 bacteria and grows by 18% each hour. Create an exponential model for the population after $t$ hours, then find how many hours it takes for the population to exceed 2000. Round to the nearest tenth of an hour and interpret your result.
7. The matrix equation $\begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\7\end{bmatrix}$ represents what system of linear equations?​
8. For $p(x)=x(x-2)^2(x+3),$ identify zeros with multiplicities, determine whether the graph crosses or touches the x-axis at each zero, and state the end behavior.
9. Consider the nested expression $H(x)=2(3x+1)^3.$ If you view $(3x+1)$ as a single entity, which statement best describes the structure and what depends on $x$?
10. Use the Rational Zeros Theorem to list all possible rational zeros of $P(x)=5x^4+2x^3-3x^2+x-4.$
11. Solve for $x$: $5\cdot 2^{3x}=60$. Express your solution as a logarithm (base 2) and then approximate using a calculator.
12. Which statement correctly explains why linear functions grow by equal differences while exponential functions grow by equal factors over equal intervals (without using calculus)?
13. Consider the functions $f(x)=5x$ (linear), $g(x)=x^2$ (quadratic), $h(x)=x^4$ (polynomial), and $p(x)=2^x$ (exponential). Order the functions by eventual growth rate (for sufficiently large $x$).
14. A theater is selling adult tickets (x) and student tickets (y). Adult tickets cost $15 and student tickets cost $9. The theater wants revenue of at least $450, but can sell at most 40 tickets total due to seating. Which system of inequalities models the constraints (including non-negativity)?
15. Restrict the domain of $f(x)=(x-3)^2+1$ (domain: all real numbers) so that it passes the horizontal line test and is invertible. Which domain restriction works?
16. On the complex plane, the point $z=a+bi$ corresponds to $(a,b)$. The distance between $z_1=a+bi$ and $z_2=c+di$ is |z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2, the same as the coordinate distance formula. What is the distance between $z_1=3+2i$ and $z_2=1-i$?
17. Use logarithms to solve $5\cdot 2^{3t}=60$ for $t$. Express your answer as a logarithm and then approximate using a calculator.
18. The compound interest formula $A = P(1 + \frac{r}{n})^{nt}$ can be rearranged to find the principal. If an investment grows to $8000 after 3 years with 6% annual interest compounded quarterly, which setup correctly finds the initial principal?
19. Which statement correctly connects function type to sequence type when sampling at integer inputs (equal intervals of $\Delta x=1$)?
20. Using the identity $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz$, if $a + b + c = 10$ and $a^2 + b^2 + c^2 = 38$, what is the value of $ab + ac + bc$?

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