Practice Test 41 - Algebra 2

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

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1. A cubic polynomial $s(x)$ has exactly two x-intercepts visible on its graph, with the graph crossing the x-axis at $x = -1$ and appearing to just touch the x-axis at $x = 3$. Which factored form best represents this polynomial?
2. Complete the square to write the circle equation $x^2 + y^2 - 6x + 8y - 11 = 0$ in standard form $(x-h)^2 + (y-k)^2 = r^2$.
3. Write matrices $A$, $X$, and $B$ so that the system can be written as $AX=B$ (use variable order $x, y, z$): $$\begin{cases} x+3y=5\\ 2x-z=1\\ y+2z=4 \end{cases}$$ (Note: include a $0$ coefficient for any missing variable in an equation.)
4. Let $f(x)=x^2$ be the parent function. What transformation changes $f(x)$ to $g(x)=f(x)+3$?
5. From the table, determine whether $y$ shows constant percent growth/decay or not. $x$: 0, 1, 2, 3 $y$: 50, 60, 72, 86.4 If it is constant percent change, what is the percent rate per 1-unit increase in $x$?
6. Convert the complex number $1 - \sqrt{3}\,i$ from rectangular form to polar form $r(\cos\theta + i\sin\theta)$, with $\theta$ in degrees. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan\left(\frac{b}{a}\right)$ with quadrant adjustment (argument measured from the positive real axis).
7. The arithmetic sequence is $8,\ 13,\ 18,\ 23,\dots$ where the first term corresponds to $x=1$. Write a linear function $f(x)$ that gives the $x$th term.
8. Over the complex numbers, the difference of squares identity $a^2-b^2=(a+b)(a-b)$ still holds. Factor the polynomial $x^2+9$ completely over $\mathbb{C}$.
9. Write both recursive and explicit formulas for the geometric sequence $80,\ 64,\ 51.2,\ 40.96,\ \dots$ (domain: positive integers).
10. A force in a plane is represented by the complex number $F=3+4i$ (real part is the horizontal component, imaginary part is the vertical component). What is the magnitude of the force, interpreted as the modulus $|F|=\sqrt{3^2+4^2}$?
11. How many real solutions (intersection points) does the linear-quadratic system have? $\begin{cases} y = x^2 + 2x - 3 \\ y = -x + 1 \end{cases}$​
12. After completing the square, the equation $2x^2 - 12x + 7 = 0$ can be written as $2(x - h)^2 = k$. Using this form, what are the solutions to the original equation?
13. All points $(x,y)$ that are exactly 6 units from the origin form a circle. What equation models this relationship, and how should the coordinate axes be labeled and scaled to graph it accurately?
14. Convert the complex number $3+4i$ from rectangular form $a+bi$ to polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan(b/a)$ (with quadrant adjustment). The argument $\theta$ is measured counterclockwise from the positive real axis, in degrees.
15. For $f(x)=(x-1)^2$ restricted to the domain $x\ge 1$, the function is invertible. What is $f^{-1}(x)$?
16. Prove by composition that $f(x)=\dfrac{x-4}{3}$ and $g(x)=3x+4$ are inverse functions. Compute $f(g(x))$ and $g(f(x))$ and simplify each.​
17. Multiply and write the result in standard form (polynomials are closed under multiplication): $(2x^3 - x + 4)(x^2 - 3)$
18. Which statement about coefficients and function type is correct for long-term behavior? Compare $f(x)=1000x^3$ (polynomial) and $g(x)=1.05^x$ (exponential).
19. A sequence can be viewed as a discrete function $f$ whose domain is the positive integers $n \in \{1,2,3,\dots\}$. For the Fibonacci sequence defined by $f(1)=f(2)=1$ and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$, what is $f(7)$?
20. A savings account is modeled by $A(t)=A_0(1.06)^t$, where $t$ is in years. Which expression is equivalent and reveals the daily growth factor (assume 365 days per year) applied over $365t$ days?

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