Practice Test 11 - Algebra 2

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

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1. A print shop charges a one-time setup fee of $80 plus $12 per T-shirt. Create an inequality for the total cost to stay within a $500 budget, and find the maximum number of T-shirts that can be ordered. Let $n$ = number of T-shirts.
2. Use polynomial division to rewrite the rational expression $\frac{2x^3+3x^2-5x+6}{x-2}$ in the form $q(x)+\frac{r(x)}{x-2}$, where $\deg(r)<\deg(x-2)$ (the polynomial division algorithm).
3. Describe the end behavior of the polynomial $p(x)=3x^5-2x^3+7x-1$ as $x \to \pm \infty$.
4. A company sells a product for $15 each. The cost to produce $q$ items is $C=7q+180$ dollars. 1) Write an equation for profit $P$ (dollars) in terms of $q$ (items). (Profit = revenue − cost.) 2) Choose appropriate axes labels and a reasonable scale to graph $P$ for $0\le q\le 60$. Which choice is correct?​
5. An exponential equation has the form $6 \cdot 2^{x-3} = 192$. Two students solve this differently. Student A simplifies to $2^{x-3} = 32$ and uses $\log_2$. Student B takes $\ln$ of the original equation immediately. Both methods are valid, but which statement correctly describes the relationship between their approaches?
6. Find the intersection point(s) of the line and the circle by solving the system. $\begin{cases} x^2 + y^2 = 25 \\ y = -3x \end{cases}$ What are all solutions $(x,y)$?​
7. The function $g(x) = |2x - 6| + 1$ is not one-to-one on its natural domain. If we restrict the domain to make $g$ invertible, and we want the resulting function to include the point $(1, 5)$, which domain restriction is appropriate?
8. Solve by completing the square: $x^2+8x+1=0$
9. An investment account starts with $800 and earns 6% interest compounded annually. Model the situation with an exponential equation and solve for the number of years $t$ it takes for the account to reach $1200. Let $t$ = number of years.
10. For the trigonometric function $p(x)=2\sin(3x)-1$, what are the amplitude, period, and midline? (Compare to the parent $y=\sin(x)$.)
11. Define $f(x)=|x-2|$ and $g(x)=x^2-4$. How many solutions does $f(x)=g(x)$ have? (You may use a graphing calculator to count intersection points.)
12. Consider the polynomial $f(x) = 4x^4 - 12x^3 + 9x^2 - 27x + 18$. A student correctly lists all possible rational zeros using the Rational Zero Theorem, then discovers that exactly half of these candidates are actually zeros of $f(x)$. How many zeros of $f(x)$ are rational?
13. Derive the quadratic formula by completing the square on the general equation $ax^2 + bx + c = 0 \quad (a \neq 0).$ Which expression for $x$ results?
14. For $f(x)=\dfrac{x^2-1}{x-1}$, does the function have a vertical asymptote at $x=1$? Identify any hole(s) and any horizontal or oblique asymptote.​
15. Use Pascal's Triangle (row 5: $1,5,10,10,5,1$) to expand $(x+2)^5.$
16. If $g(x) = x^4 - 3x^3 + 2x^2 - 5x + 1$ and the remainder when $g(x)$ is divided by $(x - c)$ is $15$, which equation must be satisfied by $c$?
17. Express $\frac{2x^2+7x-3}{x+3}$ as $q(x)+\frac{r(x)}{x+3}$ using the polynomial division algorithm, where $\deg(r)<\deg(x+3)$.
18. For $f(x)=\frac{x^2-4}{x-3},$ identify the zero(s), vertical asymptote(s), and oblique asymptote.​
19. Use logarithms to solve $5\cdot 2^{3t}=60$ for $t$. Express your answer as a logarithm (exact form) and then approximate using a calculator.
20. In the compound interest model $A = P(1+r)^n$, $P$ is the initial principal, $r$ is the interest rate per period, and $n$ is the number of periods. Interpret the expression by viewing parts as single entities: which statement correctly describes the structure of $P(1+r)^n$ as a product and identifies what does not depend on $P$?

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