Practice Test 60 - Algebra 2

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

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1. Show the symmetry type of $f(x)=x^3-x$ by identifying it as even, odd, or neither.
2. Evaluate $\log_2(32)$ using the inverse relationship $b^y=x \Leftrightarrow \log_b(x)=y$.​
3. Consider the function $g(x)=x^2+2$. Is the rate of change constant? Use equal intervals of $\Delta x = 1$ to decide.
4. Identify the perfect square trinomial structure and rewrite: $x^2 - 10x + 25$
5. A smoothie shop is mixing two smoothies: fruit smoothie and protein smoothie. Let $x$ = number of fruit smoothies and $y$ = number of protein smoothies. Constraints: - Each fruit smoothie uses 1 cup of fruit; each protein smoothie uses 2 cups of fruit; there are at most 18 cups of fruit: $x+2y\le 18$. - Each fruit smoothie uses 0 scoops of protein powder; each protein smoothie uses 1 scoop; there are at most 7 scoops: $y\le 7$. - They want to make at least 10 smoothies total: $x+y\ge 10$. - Non-negativity: $x\ge 0$, $y\ge 0$. Check if $(x,y)=(6,7)$ satisfies all constraints.
6. A streaming service charges a one-time setup fee and then a monthly charge. The total cost after $m$ months is modeled by $T = 12m + 30$, where $T$ is in dollars. Explain what the slope $m=12$ represents in this context.
7. The complex plane below shows the real axis (horizontal) and imaginary axis (vertical). Points $P$ and $Q$ are plotted at $P(3,2)$ and $Q(-1,-4)$. Which pair of complex numbers matches points $P$ and $Q$? (Recall: the point $(a,b)$ represents $a+bi$.)​
8. A video gets 300 views on day 1, and then gains 40 more views each day. Let $v(n)$ be the number of views on day $n$, with domain $n \in \{1,2,3,\dots\}$. Write both a recursive formula and an explicit formula for $v(n)$.
9. Apply the sum of cubes identity to factor the polynomial completely: $x^3+27.$
10. Solve the system graphically by finding the intersection point of the two lines. Choose the best answer (an exact ordered pair). $\begin{cases} y = x + 2\\ y = -2x + 8 \end{cases}$
11. Add and simplify to lowest terms. (Rational expressions behave like fractions: find an LCD, combine, and reduce.) $\frac{3}{x-2}+\frac{2}{x+2}$ (Assume $x \neq 2, -2$.)
12. A complex number $z$ satisfies $z^2 = -9$. If $z = a + bi$ where $a$ and $b$ are real, which statement about the possible values of $a$ and $b$ is correct?
13. A right circular cone has volume $V=\dfrac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height. Solve for $r$ in terms of $V$ and $h$.
14. Which method is most efficient for solving $(2x-1)^2=25$? A. factoring B. taking square roots C. completing the square D. quadratic formula
15. Consider the polynomials $f(x) = 2x^3 - 3x + 4$ and $g(x) = x^2 - 2x + 1$. If $h(x) = f(x) - 2g(x) + g(x) \cdot f(x)$, what is the degree of $h(x)$?
16. Use technology (calculator or software) to solve the system by the matrix inverse method. $$\begin{cases} x + 2y - z = 4 \\ 2x - y + z = 1 \\ x + y + 2z = 8 \end{cases}$$ Write the system as $AX=B$ and compute $X=A^{-1}B$. $$A=\begin{bmatrix}1&2&-1\\2&-1&1\\1&1&2\end{bmatrix},\quad X=\begin{bmatrix}x\\y\\z\end{bmatrix},\quad B=\begin{bmatrix}4\\1\\8\end{bmatrix}$$ What is $X$?
17. A student analyzes two sequences: Sequence A: 12, 17, 22, 27, 32, ... and Sequence B: 5, 15, 45, 135, 405, ... The student claims that Sequence A grows by equal differences while Sequence B grows by equal factors. To verify this claim, which calculations should be performed?
18. Derive the equation of a circle with center $(2,-3)$ and radius $4$ using the Pythagorean Theorem (distance formula).
19. Verify constant-ratio growth over an equal interval of length $h=2$ for the exponential function $g(x)=4\cdot (1.5)^x$ by simplifying $\dfrac{g(x+2)}{g(x)}$.
20. Construct an exponential function from the geometric sequence $3,\ 12,\ 48,\ 192,\ \dots$ where the first term corresponds to $x=1$ (so $f(1)=3$). Write $f(x)$ in the form $f(x)=a\cdot b^x$.

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