Practice Test 7 - Algebra 2

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

Divide and express in standard form $a + bi$ : $\frac{3 + 2i}{1 - i}$ (Use the conjugate of the denominator.)
Write the system as a matrix equation $AX=B$ (use variable order $x, y, z$ ). Remember to use coefficient $0$ for any missing variable: $\begin{cases} x+3y=5 \\ 2x-z=1 \\ y+2z=4 \end{cases}$
A radioactive sample has a half-life of 30 years and starts with 160 grams. Create an exponential model for the mass remaining after $t$ years, then find how many years it will take for the sample to decay to 10 grams. Round to the nearest tenth of a year and interpret your result.
A sequence satisfies $a_1 = 2$ and $a_{n+1} = a_n^2 - 3a_n + 4$ for $n \geq 1$ . When viewed as a function, what is $f(3)$ where $f(n) = a_n$ ?
A quadratic equation $ax^2 + bx + c = 0$ has discriminant $\Delta = 36$ . If the equation is solved using the quadratic formula, and the solutions are $x = \frac{-4 \pm 6}{10}$ , what is the original equation?
The midpoint of the segment from $z_1$ to $z_2$ on the complex plane is $\dfrac{z_1+z_2}{2}$ , which averages real parts and imaginary parts (just like the coordinate midpoint formula). What is the midpoint of the segment connecting $z_1=7-i$ and $z_2=1+3i$ ?
Based on the graph shown, which function could represent the exponential curve displayed?
Find the quotient $q(x)$ and remainder $r(x)$ when dividing $x^3+4x^2-2x+7$ by $x+2$ , and write $\frac{x^3+4x^2-2x+7}{x+2}=q(x)+\frac{r}{x+2}$ with $\deg(r)<\deg(x+2)$ .
Rewrite by viewing a repeated sub-expression as a single unit: $5(x-2)^2 + 7(x-2) - 3.$ Let $u = (x-2)$ . Which choice correctly describes the structure in terms of $u$ (without expanding back in $x$ )?
Use logarithms to solve $15\cdot 2^{2t}=500$ for $t$ . Express the solution as a logarithm (exact form) and then approximate using a calculator.
The function $f(x)=x^2-4x+3$ fails the horizontal line test on all real numbers because it is a parabola. Which restriction makes $f$ invertible?
If $z = a + bi$ where $a$ and $b$ are real, and $z \cdot \overline{z} = 25$ , and $z + \overline{z} = 6$ , what are the possible values of $z$ ?
A movie theater sells popcorn in two sizes. Small costs $4 and large costs $7. Let $x$ be the number of small popcorns, $y$ be the number of large popcorns, and $T$ be the total cost in dollars. Write an equation relating $T$ , $x$ , and $y$ . Then, to graph the relationship between $y$ and $x$ when $T=70$ , choose correct axes labels and a reasonable scale for $0\le x\le 20$ .
Use the table to answer the question. A school cafeteria plans to serve two lunch options with specific nutritional requirements. Option 1 costs $3.50 per serving and Option 2 costs $4.25 per serving. The cafeteria must serve exactly $400$ lunches daily while staying within a $1500 daily food budget. Additionally, they need to meet minimum daily nutritional targets. If $x$ represents servings of Option 1 and $y$ represents servings of Option 2, which system best models these constraints?
Graph the logarithmic function $f(x)=\ln(x)+2$ showing the vertical asymptote and the x-intercept.
Derive the equation of a circle with center $(0,0)$ and radius $3$ using the idea that the distance from $(x,y)$ to the center is $3$ .
Order the following functions by eventual growth rate (fastest to slowest): - $a(x)=0.001\cdot 3^x$ (exponential) - $b(x)=x^6$ (polynomial) - $c(x)=500x^2$ (quadratic) - $d(x)=12x$ (linear)
Show the symmetry type by identifying whether $f(x)=x^3-x$ is even, odd, or neither.
Let $f(x)=\sqrt{x}$ be the parent function. Describe the effect of $g(x)=f(2x)$ on the graph of $f$ .
Function $a(x) = 3\sin(2x) + 1$ and function $b$ is shown in the table. Compare the periods of these two functions.

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