Practice Test 4 - Algebra 2

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

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1. Apply the sum of cubes identity to factor completely: $x^3+27.$ Which factorization is correct?​
2. Function $f$ is given algebraically by $f(x) = -2(x-1)^2 + 8$. Function $g$ is described verbally as: "A parabola that opens downward with vertex at $(3, 6)$." Which function has the larger maximum value?
3. What is the LCD (least common denominator) for the rational expressions $\displaystyle \frac{3}{x^2-4}$ and $\displaystyle \frac{2}{x^2+4x+4}$? (Factoring first is recommended.)
4. In the expression $2x^3 - 5x^2 + 3x - 1$, suppose the coefficient of the $x^2$ term is changed to its opposite. What would be the new coefficient of $x^2$, and how would this affect the sum of all coefficients?
5. How does the graph behave at each zero of $g(x)=(x-3)^2(x+1)^3?$ Include multiplicities and the end behavior.
6. The function $h(x) = \frac{4x - 3}{x + 2}$ has domain $x \neq -2$. If $h^{-1}(x)$ exists, what is the expression for $h^{-1}(x)$?
7. Recognize the structure and factor completely as a difference of squares: $x^4 - 16$
8. A number and its reciprocal have a sum of $\frac{10}{3}$. What equation represents this context? Solve it and interpret the solutions. Let $x$ = the number (assume $x\ne 0$).​
9. Sketch the logarithmic function $u(x)=\log_3(x)-2$ showing the x-intercept and the vertical asymptote. Which pair is correct?​
10. Rewrite $\dfrac{(1.10)^{5t}}{(1.10)^{2t}}$ as a single exponential expression using the quotient of powers property.​​
11. Use polynomial division to rewrite the rational expression $\frac{2x^3+3x^2-5x+7}{x+2}$ in the form $q(x)+\frac{r(x)}{x+2}$, where $\deg(r)<\deg(x+2)$.
12. Based on the graph shown, which time interval demonstrates a constant rate of change in the car's position?
13. A theater sells tickets for $18 each. The theater also charges a flat online processing fee of $4 per order (no matter how many tickets). Write an equation for total cost $T$ (dollars) in terms of number of tickets $n$ (tickets), and choose appropriate axes labels and scales to graph for $0\le n\le 12$. Which choice is correct?​
14. Solve the equation by any method. If the solutions are complex, write them in $a\pm bi$ form: $3x^2+6x+5=0$
15. Use a calculator to solve the exponential equation $9\cdot e^{2t}=140$ for $t$. Express your answer in exact logarithmic form and as a decimal approximation.
16. A streaming service has 5000 subscribers and grows by 12% each month. Set up an exponential equation and solve for the number of months $t$ until it reaches 10,000 subscribers. Let $t$ = time in months.​
17. Solve using completing the square (do not factor): $x^2 + 12x + 20 = 0.$
18. For the geometric series $5+15+45+135+405,$ identify $a$, $r$, and $n$, then find $S_n$.
19. A tank is being filled so that the volume increases by 12 liters every 3 minutes. Is the rate of change of volume with respect to time constant? If so, what is the constant rate in liters per minute?
20. Are $f(x)=5-2x$ and $g(x)=\dfrac{5-x}{2}$ inverses? Verify by composition by computing $f(g(x))$ and $g(f(x))$ and checking whether both equal $x$.

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