Practice Test 63 - Algebra 2

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

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1. What function undoes $f(x)=\dfrac{x-4}{2}$? (That is, find $f^{-1}(x)$.)​
2. Solve for $x$: $12 \cdot e^{2x} = 90$. Express the solution as a logarithm and approximate using technology.
3. Solve the linear-quadratic system algebraically (by substitution). The solutions correspond to the intersection point(s) of the line and the parabola. $\begin{cases} y = x + 1 \\ y = x^2 - 2x + 1 \end{cases}$ What are all solutions $(x,y)$ to the system?
4. A radioactive sample has a half-life of 12 years and starts with 160 grams. Create an exponential model for the mass remaining after $t$ years, then find how long it will take until only 20 grams remain. Round to the nearest tenth of a year and interpret your result in context.
5. The value of a used laptop is modeled by $V(t)=900(0.78)^t$, where $t$ is the number of years since purchase and $V(t)$ is in dollars. In this function, what does it mean that the base is $0.78$?
6. On the complex plane, addition corresponds to vector addition. If $z_1=2+i$ and $z_2=1+3i$, what point represents $z_1+z_2$?​
7. Based on the table shown, determine which function type best models each dataset and identify the key growth characteristic.
8. Use the logarithm 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 the expression $\log_7\left(\frac{\sqrt{x}\,y^3}{w^2}\right).$
9. Convert the rectangular complex number $1-\sqrt{3}i$ to polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\!\left(\frac{b}{a}\right)$ with quadrant adjustment. (Angle in degrees.)
10. Solve the quadratic equation and express any complex solutions in $a \pm bi$ form: $x^2 + 6x + 13 = 0$
11. A streaming service models its subscriber count by $S(t) = 50000(1.02)^t$, where $S$ is the number of subscribers and $t$ is time in months. What is the growth rate as a percent per month?
12. A radioactive substance decays according to the model $N(t) = N_0 e^{-0.693t}$, where $t$ is in years. If $N_0 = 800$ grams and the substance decays to 50 grams, which equation correctly represents the solution for $t$?
13. The expression $6x^2 + x - 12$ can be factored by grouping or by finding two numbers that multiply to $ac = -72$ and add to $b = 1$. What are the zeros of $f(x) = 6x^2 + x - 12$?
14. A trinomial has a perfect-square pattern. Which rewrite correctly expresses $x^2-10x+25$ using structure?
15. Let $z=3+4i$ and $w=1-i$. On the complex plane, the distance between the points representing $z$ and $w$ is $|z-w|$. What is this distance?​
16. Use structure to factor the expression completely by viewing it as a difference of squares twice: $x^4 - y^4$
17. Which statement correctly explains why $f(x)=x^2$ (domain all real numbers) needs a domain restriction to have an inverse function, in terms of the horizontal line test?
18. A town's population is modeled by the function $P(t)=12000(1.03)^t$, where $t$ is time in years. What constant percent rate does this model represent, and is it growth or decay?
19. A streaming service tracks subscribers each month: Month $m$: 1, 2, 3, 4 Subscribers: 12,000; 12,600; 13,230; 13,891.5 Does the table show a constant percent growth rate? If yes, what is the percent growth rate per month?
20. To sketch $f(x)=x^2+2x-8$, first factor it to find the zeros. Which choice gives the correct factored form and x-intercepts?

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