Practice Test 103 - Algebra 2

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

Start practice test

1. Write an expression for $f^{-1}(x)$ given $f(x)=x^3+5$.​​​
2. A function $h(x)$ satisfies $h(0) = 3$, $h(2) = 15$, $h(4) = 75$, and $h(6) = 375$. A function $k(x)$ satisfies $k(1) = 8$, $k(3) = 14$, $k(5) = 20$, and $k(7) = 26$. If both functions maintain their respective growth patterns, which comparison of their values at $x = 8$ is most accurate?
3. A town has an initial population of 18,000 and grows by 2.2% per year. Create an exponential model for the population after $t$ years, then find the population after 15 years. Round to the nearest person and interpret your answer.
4. Use synthetic division to divide $P(x)=x^4-2x^3+0x^2+5x-6$ by $(x+1)=(x-(-1)).$ Set up with $c=-1$ and coefficients $1,-2,0,5,-6$ (include the $0$). Read the quotient from the bottom row and the remainder as the last number (which equals $P(-1)$). Which result is correct?
5. A bacteria culture grows according to $P(t)=500(1.12)^t$, where $P$ is the number of bacteria and $t$ is time in hours. What is the meaning of the parameter $500$ in this model?
6. A quadratic equation has the completed square form $3(x - 4)^2 = 12$. If this same equation is written in the form $(x - h)^2 = k$, what is the value of $k$?
7. What is the end behavior of $f(x)=-3x^5(x-2)^2$? (You may use only the leading term to decide.)
8. Complete the square: $x^2+y^2-4x+8y=5$. Which is the correct standard form of the circle?
9. In an AC circuit, the impedance is $Z=4+3i$ ohms. A common calculation uses $Z\overline{Z}$, which equals the modulus squared. Compute $Z\overline{Z}$.
10. Which of the following expressions is equivalent to $\overline{\left(\frac{2 + i}{1 - i}\right)}$?
11. A polynomial $g(x)$ has leading coefficient $-3$ and constant term $20$. After applying the Rational Zero Theorem and testing several candidates, a student determines that $\frac{4}{3}$ and $-\frac{5}{1}$ are zeros of $g(x)$. What is the minimum possible degree of $g(x)$?
12. For the arithmetic sequence defined as a function by $f(1)=3$ and $f(n)=3+6(n-1)$ for integers $n\ge 1$, which choice gives an equivalent recursive definition for $f(n)$?
13. Let $z = 3 - 4i$. What is the complex conjugate $\overline{z}$? (Recall: the conjugate of $a+bi$ is $a-bi$.)
14. The sequence $5,\ 15,\ 45,\ 135,\ \dots$ is geometric. Write both a recursive formula and an explicit formula for $a_n$ (with $n\ge 1$).
15. Determine whether the relationship is linear or exponential, and write the function that models it. $x$: $0,\ 1,\ 2,\ 3$ $f(x)$: $81,\ 27,\ 9,\ 3$ Which function matches the data?
16. Expand the expression using logarithm properties: $\log_{10}\left(\frac{100x^4}{y^2}\right)$ (Assume $x>0$ and $y>0$.)
17. Prove the identity $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ by expanding $ $(a-b)^3$=(a-b)(a-b)(a-b) $. Which expansion is correct?
18. A taxi fare is modeled by $F = 2.75m + 4.50$, where $F$ is the total fare in dollars and $m$ is the number of miles traveled. In this model, interpret the parameter $2.75$.
19. Use the table of values for $p(x)=x^3$ (polynomial) and $e(x)=2^x$ (exponential). Which conclusion best matches the long-term behavior shown? (Assume the pattern continues for larger $x$.)
20. Determine whether the system has one solution, no solution, or infinitely many solutions: $\begin{cases} 3x - y = 6\\ 6x - 2y = 12 \end{cases}$

Practice tests