Practice Test 49 - Algebra 2

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

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1. Solve for $x$: $\log_2(x)=4$.​
2. Factor the quadratic $x^2+25$ over the complex numbers. Recall that every quadratic $ax^2+bx+c=0$ has exactly 2 complex zeros counting multiplicity (2 distinct real, 1 repeated real, or 2 complex conjugates), and the discriminant indicates the type. Which factorization and zero count is correct?
3. Convert the system to $AX=B$ (use variable order $x, y, z$): $$\begin{cases} 5x-2y+z=0\\ -x+3y+4z=10\\ 2x+y-3z=-6 \end{cases}$$
4. A rational expression is written as $\dfrac{3x^2 - 2x + 5}{x - 1}$. Which list correctly gives the terms in the numerator?
5. Solve by factoring: $x^2-7x+10=0.$
6. Which statement about long-term growth is true as $x\to\infty$?
7. Solve using the method of completing the square: $x^2+2x+5=0.$ Express solutions in $a \pm bi$ form.
8. A quadratic equation has the form $ax^2 + bx + c = 0$ where $a = 3$, $b = -12$, and $c = k$. If the equation can be solved by factoring into the form $3(x - p)(x - q) = 0$ where $p$ and $q$ are integers, and one solution is $x = 5$, what is the value of $k$?
9. How does replacing $f(x)$ with $f(3x)$ change the graph of $f(x)$?
10. Restrict the domain of $f(x)=-x^2+10$ (domain: all reals) so that it is invertible. Which restriction works?
11. Let the parent function be $f(x)=x^3$. What is the effect on the graph of replacing $f(x)$ with $f(2x)$?
12. Verify the formula by computing both ways: Which value equals $S_4 = 3 + 6 + 12 + 24$ and also equals $\dfrac{3(1-2^4)}{1-2}$?
13. Find the sum of the first 8 terms of the geometric series with first term $a=7$ and common ratio $r=\tfrac{1}{2}$.
14. The function $f(x)=(x-3)^2+1$ (domain: all real numbers) fails the horizontal line test. What domain restriction makes $f$ invertible?
15. Solve $f(x)=27$ where $f(x)=(x+2)^3$, and express your solution as $f^{-1}(27)$.​​​
16. Factor $r(x)=4x^2-9$ to reveal its zeros. Then identify the x-intercepts of the graph.
17. Two tables describe relationships between $x$ and $y$ at equal $x$-intervals. Table 1: x: 0, 1, 2, 3; y: 6, 9, 12, 15 Table 2: x: 0, 1, 2, 3; y: 2, 4, 8, 16 Which statement is correct?
18. A function $p(x)$ satisfies $p(x+2)-p(x)=10$ for every real number $x$. Which conclusion is justified about the growth pattern and the type of sequence formed by $p(0),p(2),p(4),\dots$?
19. For $f(x)=\frac{x^2-1}{x+1},$ does the graph have a vertical asymptote at $x=-1$? Identify any hole and give its coordinate.
20. The modulus of $z=a+bi$ is $|z|=\sqrt{a^2+b^2}$. Which complex number has modulus $5$?

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