Practice Test 5 - Algebra 2

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

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1. Identify the perfect square structure and rewrite the expression as a squared binomial: $x^2 - 10x + 25$
2. The graphs of $f(x) = \log_2(x + 4)$ and $g(x) = 3 - x$ intersect at point P. Which statement best explains why the x-coordinate of point P is also a solution to $\log_2(x + 4) + x = 3$?
3. The rational function $p(x) = \frac{4x^3 + 8x^2 - 12x}{2x^2 + 2x - 12}$ simplifies after factoring. What is the oblique asymptote of the simplified function?
4. For $f(x)=\frac{(x-3)(x+1)}{(x-3)(x-2)},$ identify the zero(s), vertical asymptote(s), and any hole (removable discontinuity).
5. In the expression $3(2x + 1)^2$, what is the coefficient of $(2x + 1)^2$?
6. Which function eventually grows fastest as $x\to\infty$?​
7. Solve by completing the square: $x^2+2x+5=0$. (Give solutions in $a\pm bi$ form.)
8. A culture of bacteria triples every day. If $t$ is measured in weeks, its growth factor can be written as $3^{7t}$ (since 7 days per week). Rewrite $3^{7t}$ using exponent properties as a power of a power so it looks like a weekly growth factor raised to $t$. Which rewrite is correct?
9. A medication dose begins at 300 mg in the bloodstream. Each hour, 20% of the medication is eliminated (so 80% remains). Set up an exponential equation and solve for the time $t$ (in hours) when 90 mg remains. Let $t$ = time in hours.
10. 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 as a sum and/or difference of logarithms: $$\log_3\left(\frac{x^2y}{z^3}\right).$$
11. Let the parent function be $f(x)=\sqrt{x}$. How does the graph change when you replace $f(x)$ with $g(x)=f\!\left(\tfrac12 x\right)$?​
12. For the geometric sequence $50,\ 35,\ 24.5,\ 17.15,\ \dots$ (as a function with integer domain $n\ge 1$), write both a recursive definition and an explicit formula for $a_n$.
13. Subtract and simplify to lowest terms: $\frac{x+1}{x^2-4}-\frac{3}{x-2}$ (Assume $x \neq 2, -2$.)
14. A company tracks the number of active users each month: Month: 0, 1, 2, 3 Users: 12,000; 13,200; 14,520; 15,972 Does this represent constant percent change, and if so, what is the percent growth rate per month?
15. Treat $z=a+bi$ as the point $(a,b)$ on the complex plane. Using the midpoint formula $\frac{z_1+z_2}{2}=\frac{a+c}{2}+\frac{b+d}{2}i,$ what is the midpoint of the segment joining $z_1=-1+3i$ and $z_2=5+i$?
16. Determine whether the function $h(x)=x^4-2x^2$ is even, odd, or neither.
17. Determine whether the relationship is linear or exponential, then write the function. The table gives: $x$: 1, 2, 3, 4 $y$: 2, 6, 18, 54 Which function models the data?
18. Use structure to show two levels of factoring are possible. Factor completely: $x^4 - 1$
19. If $f(x) = x^3 + 8$ and $g(x) = x + 2$, then $\frac{f(x)}{g(x)}$ can be written as a polynomial for all values of $x$ except $x = -2$. What is this polynomial?
20. Describe the end behavior of the exponential function $f(x)=5\cdot (0.6)^x$.

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