Practice Test 6 - Algebra 2

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

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1. Solve by completing the square: $x^2 - 8x + 5 = 0$ Write it in the form $(x - p)^2 = q$ and solve for $x$.
2. A taxi company charges a flat pickup fee plus a per-mile charge. The total fare is modeled by $F = 2.75m + 4.50$, where $m$ is the number of miles traveled. What does the parameter $4.50$ represent in this context?
3. Use a calculator to solve the exponential equation $4\cdot 2^{t/5}=30$. Give the exact solution as a logarithm and an approximate decimal value.
4. Find the inverse function $f^{-1}(x)$ for $f(x)=2x^3$.
5. A student claims that $f(x)=x^2$ (with domain restricted to $x\ge 0$) and $g(x)=\sqrt{x}$ are inverse functions. Verify or disprove by computing $f(g(x))$ and $g(f(x))$ using the stated domain restriction.
6. Rewrite the circle equation $x^2+y^2+6x-4y+9=0$ in standard form $(x-h)^2+(y-k)^2=r^2$.
7. A population of rabbits grows according to an exponential model. The population triples every 4 months. If there are currently 20 rabbits, which function gives the population after $m$ months?
8. Multiply and write the result in standard form. Since polynomials are closed under multiplication, the product is a polynomial: $\left(x^2 + 2x - 5\right) \times \left(x^2 - x + 1\right).$
9. Verify or disprove: $f(x)=\dfrac{x+4}{2}$ and $g(x)=2x+4$ are inverses. Use composition to compute $f(g(x))$ and $g(f(x))$ and check whether both equal $x$.
10. A bakery models the total cost (in dollars) to make $x$ batches of muffins by $C(x)=45+12x.$ The bakery can make at most 20 batches in a day, and $x$ represents the number of batches made. What is an appropriate realistic domain for $C$?
11. Verify that the polynomial identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ is true for all real $a,b$ by expanding the right-hand side.
12. A polynomial function has zeros at $x = -2$ (multiplicity 2), $x = 1$ (multiplicity 1), and $x = 4$ (multiplicity 3). If the leading coefficient is positive, which interval contains a point where the function value is negative?
13. Solve by taking square roots: $(x-3)^2 = 5$.
14. Convert $\sqrt{2}\big(\cos 45^\circ+i\sin 45^\circ\big)$ from polar form to rectangular form $a+bi$. Use $a=r\cos\theta$ and $b=r\sin\theta$.
15. A recycling bin contains 500 pounds of material. Each day, 6% of the material is removed. Which statement correctly identifies the constant percent rate and the exponential model for the amount remaining after $d$ days?
16. If the remainder when $p(x)=x^4+mx^2-2x+1$ is divided by $(x-2)$ is $9$, what is $m$? (Use: remainder $=p(2)$.)
17. A function is defined by $f(x)=x^4-3x^2$. Does $f$ have symmetry? What type?
18. What is the 4th term (counting from the first term as 1st) in the expansion of $(x+y)^6?$
19. A student deposits $40 into a jar each week and does not remove any money. Let $a_n$ be the total amount in the jar after $n$ weeks, with $a_1=40$. Write both recursive and explicit formulas (domain: positive integers).
20. What is $f^{-1}(x)$ if $f(x)=\dfrac{x}{x+2}$, where $x\ne -2$?

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