Practice Test 68 - Algebra 2

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

Expand the expression using logarithm properties (product, quotient, power): $\log_3\left(\frac{x^2 y}{z^3}\right)$ Use: - $\log_b(xy)=\log_b(x)+\log_b(y)$ - $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$ - $\log_b(x^p)=p\log_b(x)$
Use the discriminant to determine the solution type for $x^2-4x+5=0$ , then find all solutions in the complex numbers. Remember: every quadratic $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity (2 distinct real, 1 repeated real counted twice, or 2 complex conjugates).
A ball is dropped from 10 ft. Each bounce reaches $80\%$ of the previous height. What is the total vertical distance traveled during the first 5 bounces (up-and-down for each bounce), not counting the initial drop? Hint: The bounce heights form a geometric sequence with first bounce height $10(0.8)$ .
A gym membership fee changes over time according to the table. Month $m$ : 0, 1, 2, 3 Fee $F(m)$ : 40, 44, 48, 52 Classify the change as exponential growth, exponential decay, or neither.
Which structural observation about $4x^2 - 12x + 9 - y^2$ leads to the most efficient factoring approach?
Which situation has a constant rate of change (and therefore can be modeled by a linear function)? (a) A tank fills 4 gallons every minute. (b) The area of a square as its side length increases. (c) A bacteria population doubles every hour.
A town has an initial population of 18,000 and grows by $2.2\%$ per year. Create an exponential model for the population after $t$ years, then determine the population after 15 years. Round to the nearest person and interpret your result in context.
Divide and simplify to lowest terms. This illustrates that rational expressions are closed under division by a nonzero rational expression (like rational numbers). Divide: $\frac{x^2-4}{x^2-1}\div\frac{x-2}{x+1}$ (Assume all denominators are nonzero and you are not dividing by zero.)
A cubic polynomial $f(x)$ has leading coefficient 1 and satisfies $f(1) = f(3) = f(5) = 0$ . When $f(x)$ is divided by $(x - 7)$ , the remainder is:
The expression $4\sin^3(2x) - 6\sin(2x)\cos^2(2x)$ can be restructured by viewing certain parts as single entities. Which approach most effectively reveals a factored form?
For the logarithmic function $g(x)=\log_2(x-3)$ , find the vertical asymptote and the x-intercept. (Connect to the parent function $y=\log_2(x)$ .)
Prove the identity $(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ by expanding $(a-b)^3=(a-b)(a-b)(a-b)$ . Which expansion is correct?
Let $f(x)=x^2$ be the parent function. Which description matches $g(x)=2f(x-1)+3$ ?
A small theater has 120 seats. The revenue (in dollars) from selling $n$ tickets is $R(n)=15n$ . Tickets are sold to whole people, and it is possible that zero tickets are sold. What is an appropriate domain for $n$ ?
A function has values at consecutive integers: $H(0)=5$ , $H(1)=10$ , $H(2)=20$ , $H(3)=40$ . Based on growth over equal intervals ( $\Delta x=1$ ), does this represent linear or exponential growth?
Subtract and write the result in standard form (polynomials are closed under subtraction): $(4x^4 - 2x^2 + x - 9) - (x^4 + 3x^3 - 2x + 5)$
What is the end behavior of $p(x)=3(x-2)(x+1)(x-4)^2?$ (You may also use the degree and leading coefficient to decide.)
Sketch $f(x)=2^x$ and $g(x)=\log_2(x)$ on the same axes. How are these graphs related? (Connect to the parent functions and inverse relationships.)
For $f(x)=(x-1)^2$ with restricted domain $x \ge 1$ (so it passes the horizontal line test), what is $f^{-1}(x)$ ?
A car loses $12\%$ of its value each year. If its value is $18{,}000$ dollars at year 0, let $V(n)$ be its value after $n$ years (domain: nonnegative integers). Which choice gives both a correct recursive and explicit model for $V(n)$ ?

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