Practice Test 20 - Algebra 2

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

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1. Use the properties $\log_b(xy)=\log_b(x)+\log_b(y),\quad \log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y),\quad \log_b(x^p)=p\log_b(x)$ to expand $\log\left(\frac{(mn^3)^2}{p}\right)$, where $\log$ is base 10. Write your answer as a sum/difference of logs with coefficients.
2. Restrict the domain of $f(x)=x^2$ to $x\ge 0$ so that it passes the horizontal line test. After this restriction, what is the inverse function $f^{-1}(x)$?
3. In algebra-based modeling, suppose $y = \dfrac{kx}{x+a}$ where $a$ and $k$ are constants and $x\neq -a$. Solve $y = \dfrac{kx}{x+a}$ for $x$ in terms of $y, a, k$.
4. Use synthetic division (instead of long division) to divide $P(x)=x^4-2x^3+3x^2-4x+5$ by $(x-1)$. Use $c=1$ and coefficients $1,-2,3,-4,5$. Read the quotient from the bottom row (all but the last number) and the remainder from the last number (equal to $P(1)$). Which is correct?
5. Convert the complex number $1-\sqrt{3}i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$ with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment. (Argument is measured from the positive real axis.)
6. Show that any exponential function $g(x)=a\cdot b^x$ (with $a\neq 0$ and $b>0$) grows by equal factors over equal intervals: prove that for any real number $h$, the ratio $\dfrac{g(x+h)}{g(x)}$ is constant (independent of $x$). Which expression correctly simplifies $\dfrac{g(x+h)}{g(x)}$?
7. The decibel level of a sound is modeled by $\text{dB}=10\log_{10}\left(\dfrac{I}{I_0}\right)$, where $I$ is intensity and $I_0$ is a reference intensity. A sound has intensity $2500I_0$. What is its decibel level? Round to the nearest tenth of a decibel.
8. Add and simplify to lowest terms. (Like fraction arithmetic, you must use an LCD; the result is still a rational expression.) Add: $\dfrac{x}{x-1}+\dfrac{2}{x+1}$ (assume denominators are nonzero).​
9. Write the exponential function $g(x)=a\cdot b^x$ that passes through $(0,5)$ and $(3,40)$. Which formula is correct?
10. Verify that $(x-3)$ is a factor of $p(x)=x^3-6x^2+11x-6$ using the Remainder Theorem.
11. A student graphs $y=f(x)$ and $y=g(x)$ and sees they intersect at the point $(4,7)$. Which equation must be true?​
12. Convert between exponential and logarithmic forms: Rewrite $\log_2(16)=4$ as an exponential equation.
13. Describe the end behavior of the polynomial $h(x)= -3x^5+2x^3-7x+1$ as $x\to\pm\infty$.
14. A tech company's revenue is modeled by $R(t)=R_0\,8^t$, where $t$ is in years. Rewrite $8^t$ in terms of base $2$. (Use the power-of-power property.)
15. Consider the function $f(x) = 4x - 7$. Is the rate of change constant? If so, what is the constant rate $\Delta f/\Delta x$?
16. Evaluate $\log_2(8)$ using the definition $\log_b(x)=y \Leftrightarrow b^y=x$.​
17. A complex number $z=a+bi$ is plotted as the point $(a,b)$ on the complex plane. What is the modulus of $z=6-8i$ (its distance from the origin), given by $|z|=\sqrt{a^2+b^2}$?​
18. Prove that any linear function $f(x)=mx+b$ grows by equal differences over equal intervals. Specifically, show that for any real number $h$, the difference $f(x+h)-f(x)$ is constant (does not depend on $x$), and state what constant it equals. Also connect this to why the values $f(0),f(1),f(2),\dots$ form an arithmetic sequence.
19. Solve the equation by completing the square: $x^2 + 4x + 13 = 0$ Express solutions in the form $a \pm bi$.
20. Solve by completing the square: $x^2-12x+20=0.$

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