Practice Test 66 - Algebra 2

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

Two points on the complex plane are $z_1=-4+3i$ and $z_2=2+3i$ . Using $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ , what is the distance between them? Give your answer in simplest form.
Explain domain restriction through the horizontal line test: For $f(x)=x^2$ , which statement correctly describes why a restriction is needed for an inverse?
A savings plan deposits $200 at the end of each month into an account that earns $1\%$ interest per month. If the account starts at $0, what is the total amount in the account immediately after the 6th deposit (ignore any fees)? Model the balance as a finite geometric series: $200\big(1 + 1.01 + 1.01^2 + \cdots + 1.01^5\big).$
On the same axes, $y=2^x$ and $y=\log_2(x)$ are graphed. How are these two graphs related?
Use logarithm properties to evaluate without a calculator: $\log_3\left(\frac{81}{9}\right)$
What are the factors of the term $6x^2y$ ? (Factors are parts multiplied together.)
A toy rocket's height (in feet) after $t$ seconds is modeled by $h(t)=-16t^2+64t$ . The rocket is in the air from launch until it hits the ground again at $t=4$ . What is an appropriate realistic domain for $h(t)$ ?
For the exponential function $g(x)=2\cdot 3^x$ , verify over equal unit intervals ( $h=1$ ) that the ratios are constant by computing $g(0), g(1), g(2), g(3)$ and the consecutive ratios. Which option correctly demonstrates the constant-ratio property and identifies the associated sequence type at integer inputs?
Divide and rewrite $\frac{3x^3 - x^2 + 4x - 6}{x + 1}$ in the form $q(x) + \frac{r(x)}{x+1}$ , with $\deg(r) < 1$ .
Consider the quadratic equation $x^2-6x+13=0$ (a special case of $ax^2+bx+c=0$ with real coefficients). Quadratics always have exactly 2 solutions in the complex numbers counting multiplicity: either 2 distinct real solutions ( $b^2-4ac>0$ ), 1 repeated real solution counted twice ( $b^2-4ac=0$ ), or 2 complex conjugate solutions ( $b^2-4ac<0$ ). Using the discriminant to determine the type, how many complex solutions does this equation have counting multiplicity, and what are they?
The value $V$ of an investment (in dollars) after $t$ years is given by $V(t) = 5000(1.06)^t$ . An investor wants to know when the investment will be worth at least $8,000. Which inequality should be solved?
Let $f(x)=|x|$ be the parent function. Describe the transformation from $f$ to $g(x)=\tfrac{1}{2}\,f(x)$ .
Find the modulus (absolute value) of the complex number $3 + 4i$ . (Recall $|a+bi|=\sqrt{a^2+b^2}$ .)
For the polynomial $p(x)=2(x+3)(x-1)^2(x-4)$ , identify all zeros with their multiplicities, determine whether the graph crosses or touches the $x$ -axis at each zero, and state the end behavior.
Given that $W(x) = x^5 - 2x^4 + 3x^3 - x^2 + 4x - 6$ and the remainder when $W(x)$ is divided by $(x^2 - 4)$ is $ax + b$ , what is the value of $a + b$ ?
Which statement correctly describes the key features of the exponential function $f(x)=\left(\tfrac{1}{3}\right)^x$ ?
Find the quotient $q(x)$ and remainder $r(x)$ when dividing $x^3 + 4x^2 - 2x + 7$ by $x + 2$ , and write $\frac{x^3 + 4x^2 - 2x + 7}{x + 2} = q(x) + \frac{r(x)}{x+2}$ with $\deg(r) < 1$ .
A car worth $\$18{,}000$ depreciates by 10% each year. Let $a_n$ be the value after $n$ years, with $a_1$ the value after 1 year. Write both recursive and explicit formulas (domain: positive integers).
Use Pascal's Triangle (row 4: $1,4,6,4,1$ ) to expand $(x+y)^4.$
Consider the quadratic equation $x^2-6x+13=0$ (a special case of $ax^2+bx+c=0$ with real coefficients). In the complex numbers, every quadratic has exactly 2 solutions counting multiplicity: either two distinct real solutions ( $b^2-4ac>0$ ), one repeated real solution of multiplicity 2 ( $b^2-4ac=0$ ), or two complex conjugate solutions ( $b^2-4ac<0$ ). Use the discriminant $b^2-4ac$ to determine the type of solutions and then find all complex solutions of this equation.

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