Practice Test 70 - 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 Pythagorean Theorem (distance formula) to derive the equation of the circle with center $(2,-3)$ and radius $4$. What is the equation in standard form $(x-h)^2+(y-k)^2=r^2$?
2. Solve the system graphically (estimate from the graph). The lines are $y = -x + 4 \quad \text{and} \quad y = \tfrac{1}{2}x + 1.$ What is the approximate intersection point $(x, y)$?
3. Use polynomial long division to write $\frac{x^4-3x^2+5}{x^2+1}$ in the form $q(x)+\frac{r(x)}{x^2+1}$, where $\deg(r)<2$.
4. Let $f(x)=\ln(x)$ and $g(x)=x-2$. Use a graphing calculator (or graphing software) to approximate the solution to $\ln(x)=x-2$. Give the solution to the nearest hundredth.
5. Convert between forms: Which logarithmic equation is equivalent to the exponential equation $2^7=128$?​
6. A polynomial has zeros at $x = -3, 1, 1, 4$ (listed with repetition according to multiplicity). If this polynomial has a negative leading coefficient, in which interval must the function have a positive value?
7. Which statement correctly compares the effects of $f(2x)$ and $2f(x)$ on the graph of a parent function $f(x)$?
8. A radioactive isotope decays such that 12.5% of the original amount remains after 30 days. A chemist needs to determine the daily decay rate to predict when only 1% will remain. What is the constant daily percent decay rate?
9. The sequence $2, 5, 8, 11, \dots$ is a function with domain $n \in \{1,2,3,\dots\}$. Which recursive definition matches this sequence?
10. A student is choosing smoothies. Let $x$ = number of protein smoothies and $y$ = number of fruit smoothies. Constraints: - Protein requirement: at least 30 g total. Protein smoothie has 12 g; fruit smoothie has 6 g. - Calorie limit: at most 500 calories total. Protein smoothie has 220 cal; fruit smoothie has 140 cal. - Nonnegativity: $x\ge 0,\ y\ge 0$. Determine which constraint the point $(x,y)=(1,4)$ violates (if any).
11. A bacteria culture starts with 200 bacteria and doubles every 3 hours. Create an exponential model for the population $P(t)$ after $t$ hours, then solve for when the population first reaches 10,000 bacteria. Round to the nearest tenth of an hour and interpret your result.​
12. Find the sum of the first 8 terms of the geometric series $7 + 14 + 28 + \cdots$ (8 terms total).
13. A student is choosing how many hours to spend on math (x) and history (y) studying this weekend. They have at most 9 hours total: $x+y\le 9$. They want at least 4 hours of math: $x\ge 4$. They also want history time to be at least half of math time: $y\ge \tfrac12 x$. Non-negativity applies. Which point is viable?
14. Which method is most efficient for solving the equation $x^2+5x+6=0$ and why?
15. A town has a population of 50,000 and grows by 3% each year. Model the population after $n$ years with both a recursive and an explicit formula, where $a_1$ is the population after 1 year (domain: positive integers).
16. The sequence is defined recursively by $a_1=6$ and $a_{n+1}=\tfrac{1}{2}a_n$ for $n\ge 1$. Write the explicit formula for $a_n$ (with $n$ a positive integer).
17. A student claims the system below can be written as $$\begin{bmatrix}1&-2&3\\0&4&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}7\\5\end{bmatrix}.$$ Which system does this matrix equation represent (use variable order $x, y, z$)?
18. A bacteria culture is modeled by $P(t)=200\cdot 2^{t/6}$. When does it reach $P=1500$? Solve for $t$ (in hours) using logarithms, and approximate using technology.
19. A population model is given by $P(x)=-2x^3+5x^2+1$, where $x$ is years. Describe the end behavior of $P$.
20. Use the identity $(x^2+y^2)^2=(x^2-y^2)^2+(2xy)^2$ to generate a Pythagorean triple when $x=4$ and $y=1$.

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