Practice Test 2 - Algebra 2

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

Express using the change-of-base formula (in terms of natural logs): $\log_5(20)$
Use Pascal's Triangle (row 3: $1,3,3,1$ ) to expand $(a+b)^3.$
A bakery models the number of cupcakes it can make in a day by $P(h)=24h$ , where $h$ is the number of hours worked. The bakery is open at most 10 hours per day, and $h$ can include partial hours (for example, 6.5 hours). What is an appropriate realistic domain for $h$ ?
The function $q(x) = \frac{3x - 4}{x + 1}$ has an inverse function. If $q^{-1}(2) = k$ , what is the value of $k$ ?
A small bakery makes muffins (x) and cookies (y). Each muffin uses 2 cups of flour and each cookie uses 1 cup of flour; there are at most 50 cups of flour available: $2x+y\le 50$ . The bakery must make at least 10 muffins: $x\ge 10$ . Muffins and cookies are discrete items, so $x$ and $y$ must be integers, and $x,y\ge 0$ . Is $(x,y)=(12.5,25)$ viable?
A physics model gives the height (in meters) as $h(t)=h_0+v_0t-\frac{1}{2}gt^2,$ where $h_0$ is initial height, $v_0$ is initial velocity, and $g$ is the gravitational constant. If you chunk the last two terms as a single entity $t\left(v_0-\frac{1}{2}gt\right)$ , which interpretation is correct?
In the expansion of $(x+y)^6$ , what is the coefficient of $x^3 y^3$ (use Pascal's Triangle row 6)?
Let $z_1 = 6 - 2i$ and $z_2 = 1 + 5i$ . What is $z_1 - z_2$ expressed in standard form $a + bi$ ?
Why do linear functions grow by equal differences while exponential functions grow by equal factors over equal intervals? Choose the statement that correctly describes both using $h$ as a fixed interval length.
Solve for $x$ using the inverse relationship: $\log_2(x)=6$ .
For $f(x)=\frac{x^2-1}{x-1}$ does the graph have a vertical asymptote at $x=1$ ? Identify any hole and any zero(s).
A catering service offers two meal packages for events. Package A costs $18 per person and includes $2$ appetizers and $1$ entrée per person. Package B costs $25 per person and includes $3$ appetizers and $1$ entrée per person. For an event with $n$ people, the host wants to spend at most $1200 total and serve at least $150$ appetizers. If $a$ people choose Package A and $b$ people choose Package B, which system correctly models these constraints?
For the series $5+15+45+135+405,$ identify $a$ , $r$ , and $n$ , then find the sum $S_n$ .
A student claims $f(x)=x^2$ is invertible on all real numbers because you can "solve" $y=x^2$ by writing $x=\pm\sqrt{y}$ . Which domain restriction correctly fixes the issue so the inverse is a function (passes the horizontal line test)?
Let $z_1=-2-i$ and $z_2=3+2i$ represent points on the complex plane. Using $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ (the same as the 2D distance formula), what is the distance from $z_1$ to $z_2$ ?
A polynomial function has been factored as $r(x) = (x + 1)^3(x - 2)(x - 5)^2$ . If a student incorrectly identifies the end behavior of this function, which misconception most likely led to their error?
Use the inverse relationship to evaluate $10^{\log(25)}$ , where $\log(x)$ means $\log_{10}(x)$ .
Add the polynomials (combine like terms). The sum should be a polynomial by closure under addition: $(4x^3-2x^2+6)+(-x^3+5x-9).$
A rectangular patio has an area of $96\text{ m}^2$ . Its length is 4 meters more than its width. Set up a quadratic equation and solve to find the patio's width and length. Let $w$ = the width in meters.
Complete the square: $x^2 + y^2 + 4x - 10y + 13 = 0$ . Which standard form equation is correct?

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