Practice Test 37 - Algebra 2

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

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1. A car rental company charges a daily rate plus a fixed service fee. The total cost is modeled by $C = 42d + 18$, where $d$ is the number of days rented. In this model, what does the parameter $18$ represent?
2. Which identity allows you to factor the expression $a^2-b^2$ as a product of two linear factors?
3. Recognize a difference of cubes structure and factor: $x^3 - 27$
4. Use zeros to construct a rough graph of the polynomial $p(x)=(x+2)(x-1)^2(x-4).$ Identify the zeros (with multiplicities), state whether the graph crosses or touches the x-axis at each zero, and determine the end behavior.
5. If $i^n = -1$ where $n$ is a positive integer, which statement about $n$ must be true?
6. Find $x$ in $4\cdot 2^{x/3}=30$. Express the solution as a logarithm (base 2) and approximate using technology.​​​
7. Over the real numbers, a quadratic $ax^2+bx+c=0$ can have 0, 1, or 2 real solutions. Over the complex numbers, it always has exactly 2 solutions counting multiplicity. The discriminant $b^2-4ac$ determines the solution type: $>0$ two distinct real, $=0$ one repeated real (multiplicity 2), $<0$ two complex conjugates. For $x^2+6x+9=0$, which statement is correct?
8. Multiply and simplify using $i^2 = -1$: $(3 + 2i)(1 - 4i)$. Express your answer in standard form $a + bi$.
9. Write an expression for $f^{-1}(x)$ given $f(x)=3x-7$.
10. Graph $p(x)=x^5-5x^3+4x$ by finding its zeros (from factoring) and using end behavior to make a rough sketch.
11. Let the parent function be $f(x)=\frac{1}{x}$. Describe the transformations that change $f(x)$ to $g(x)=f(x)-2.$
12. Find the modulus $r$ and argument $\theta$ (in degrees) of $3-3i$, then write it in polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment.
13. Complete the square to rewrite the expression in vertex form (a perfect square plus a constant): $x^2+8x+20$
14. Over the real numbers, $x^2+9$ does not factor. Over the complex numbers, use the difference of squares identity $u^2-v^2=(u+v)(u-v)$ by writing $x^2+9=x^2-(3i)^2$. Which is the correct factorization over $\mathbb{C}$?
15. Use log properties to expand: $\log_7\left(\frac{(xy^2)^3}{\sqrt{z}}\right).$ Write your answer as a sum/difference of logs with coefficients.
16. For the quadratic equation $x^2+4x+4=0$ (a case of $ax^2+bx+c=0$ with real coefficients), remember: every quadratic has exactly 2 complex solutions counting multiplicity. The discriminant $b^2-4ac$ determines whether the solutions are 2 distinct real, 1 repeated real (counted twice), or 2 complex conjugates. Which statement correctly describes the solutions (including multiplicity)?
17. Marcus is solving the system $\begin{cases} 5x + 2y = 19 \\ 3x - 4y = -13 \end{cases}$ using elimination. He decides to eliminate $y$ by multiplying the first equation by 2. What should he multiply the second equation by to successfully eliminate $y$?
18. A data set shows the balance in an account at the end of each quarter: Quarter: 0, 1, 2, 3 Balance ($): 2,000; 2,160; 2,332.8; 2,519.424 From the table, what is the percent growth rate per quarter?
19. Write the system with missing variables as a matrix equation $AX=B$ (use variable order $x, y, z$): $$\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$$
20. A complex number $z$ satisfies $|z| = 4$. On the complex plane (real axis horizontal, imaginary axis vertical), what set of points represents all such $z$?

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