Practice Test 45 - Algebra 2

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

Factor and graph $p(x)=x^4-5x^2+4$ showing all zeros, whether the graph crosses or touches at each zero, and the end behavior.
A quadratic polynomial $P(x) = x^2 + bx + c$ has the property that both roots lie on the circle $|z| = 3$ in the complex plane. If one root is $z_1 = -1 + 2\sqrt{2}i$ , what is the value of $c$ ?
In kinematics, the displacement of an object under constant acceleration is modeled by $s = ut + \tfrac{1}{2}at^2$ , where $s$ is displacement, $u$ is initial velocity, $a$ is acceleration, and $t$ is time. Solve for $t$ in terms of $s$ , $u$ , and $a$ .
The sequence is defined recursively by $a_1=5$ and $a_{n+1}=a_n+8$ for $n\ge 1$ . Write the explicit formula for $a_n$ (domain: positive integers).
A function is sampled at integer inputs, producing the sequence $5, 8, 11, 14, \dots$ . Which statement best classifies the growth and connects it to a function type?
A complex number is $z=6- i$ . Which expression equals $|z|^2$ (the modulus squared) using conjugates?
Complete the square to solve: $x^2 + 6x - 2 = 0.$ Write the equation in the form $(x - p)^2 = q$ and then give the solutions.
A water tank drains according to $V(t) = 500e^{-0.1t}$ gallons after $t$ minutes. The tank is considered empty when it contains less than 1 gallon. What is the appropriate domain for this function in the context of the draining process?
A compound interest problem leads to the equation $1000 \cdot e^{0.08t} = 2500$ . A student correctly simplifies this to $e^{0.08t} = 2.5$ and takes the natural logarithm to get $0.08t = \ln(2.5)$ . Using technology, $\ln(2.5) \approx 0.916$ . What is the approximate value of $t$ ?
Use the identity $(x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2$ to generate a Pythagorean triple when $x=3$ and $y=2$ . (A Pythagorean triple is $(a,b,c)$ with $a^2+b^2=c^2$ .)
Use Pascal's Triangle (row 5: $1,5,10,10,5,1$ ) to expand $(x+y)^5$ .
A company's revenue model shows that for every 50 additional customers gained, monthly revenue increases by $2,000. Currently, with 300 customers, the company earns $15,000 per month. Which statement best describes this situation?
Use technology to solve $AX=B$ by computing $X=A^{-1}B$ . $A=\begin{bmatrix}1&2&0\\0&1&3\\2&0&1\end{bmatrix},\quad B=\begin{bmatrix}5\\11\\5\end{bmatrix}$ What is $X$ ?
Solve $15\cdot e^{2t}=500$ for $t$ . Express the solution as a logarithm and approximate using technology.
Use a calculator to solve $9e^{0.4t}=200$ . Write the exact solution using a logarithm and then approximate.
An account grows by 12% per year, so its value after $t$ years is multiplied by $(1.12)^t$ . Rewrite $(1.12)^t$ in an equivalent form that reveals the monthly growth factor (assume 12 months per year).
Let $f(x)=x^3$ (the standard cubic). The function $g(x)=f(2x)$ is obtained from $f$ by which transformation?
Compute the product of a complex number and its conjugate: $(5+2i)(5-2i).$ (Recall $(a+bi)(a-bi)=a^2+b^2$ , which is always real and positive.) Give your answer in simplest form.
A bike rental shop charges a one-time $6 unlock fee plus $3 per hour. Let $C$ be the total cost (dollars) and $h$ be the number of hours rented. Write an equation for the relationship in terms of $h$ , and choose appropriate axes labels and a reasonable scale to graph the relationship for $0\le h\le 8$ .
Use technology to compute $A^{-1}$ and solve $AX=B$ using $X=A^{-1}B$ . $ A=\begin{bmatrix}4&0&1\\2&1&0\\0&3&1\end{bmatrix},\quad B=\begin{bmatrix}9\\5\\9\end{bmatrix} $ What is $X$ ?

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