Practice Test 27 - Algebra 2

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

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1. Solve $f(x)=16$ where $f(x)=2x^3$, and express your answer as $f^{-1}(16)$.​
2. A cell tower is located at $(3,4)$ on a coordinate grid. Its signal reaches all points within $10$ units of the tower. Write the equation of the circle representing points exactly $10$ units from the tower.
3. Simplify and write in standard form $a+bi$: $\frac{2-i}{3+2i}.$
4. An investment account starts with $6000 and earns $5\%$ interest compounded annually. Create an exponential model for the balance $A(t)$ after $t$ years. Then find the balance after 8 years, rounding to the nearest cent, and interpret your result in context.
5. A company models its monthly profit $P(t) = -2t^2 + 24t - 40$ thousand dollars, where $t$ represents months since opening. The company will operate as long as it remains profitable. What is the most appropriate domain for this function in the context of the problem?
6. Complete the square on the general quadratic equation to derive the quadratic formula. Starting from $ax^2+bx+c=0 \quad (a\ne 0),$ which expression for $x$ results from completing the square?
7. A quadratic with real coefficients (of the form $ax^2+bx+c=0$) always has exactly 2 complex solutions counting multiplicity. If its discriminant is negative, the solutions are non-real and come in complex conjugate pairs. If $2+3i$ is a solution of such a quadratic, what must the other solution be?
8. A toy rocket's height (in feet) after $t$ seconds is modeled by $h(t)=-16t^2+64t$. In the real situation, the rocket is launched from the ground at $t=0$ and returns to the ground when $h(t)=0$ again. What is an appropriate realistic domain for $t$?​
9. To highlight structure within complexity, consider $G = (x^2-4)\,(2x+1).$ If you chunk $(2x+1)$ as one entity, which statement correctly describes how $G$ changes when $(2x+1)$ changes, without needing to fully expand?
10. What is $f^{-1}(x)$ if $f(x)=3x-7$?​
11. How many real solutions does this linear-quadratic system have? (Solutions correspond to intersection point(s).) $\begin{cases} y = x^2 + 2x - 3 \\ y = -x + 1 \end{cases}$
12. Two circles intersect at points $A$ and $B$. The first circle has center $(1, 2)$ and passes through $(4, 6)$. The second circle has center $(3, 0)$ and the same radius as the first. What is the equation of the line passing through the intersection points $A$ and $B$?
13. Complete the square to rewrite $x^2+8x+20$ in the form $(x+a)^2+b$.
14. Let the parent function be $f(x)=|x|$. How does the graph change when you replace $f(x)$ with $g(x)=2\,f(x)$?​
15. Graph the exponential function $f(x)=3\cdot 2^x-1$ showing the y-intercept, the horizontal asymptote, and the end behavior as $x\to\infty$ and $x\to -\infty$. (Relate your sketch to the parent function $y=2^x$.)
16. A taxi company charges according to $C = 2.75x + 4.50$, where $C$ is the fare in dollars and $x$ is the distance traveled in miles. What does the parameter $2.75$ represent in context?
17. Subtract and write the result in standard form. This demonstrates closure under subtraction because the difference of two polynomials is a polynomial: $\left(x^3 + 5x^2 - 2x + 7\right) - \left(2x^3 - x^2 + 4x - 1\right).$
18. Which expression is equivalent to $\log_5\left(\frac{125}{x^3}\right)$ when $x > 0$?
19. Which set of points on the complex plane represents all complex numbers $z$ such that $|z|=5$? (Recall $|z|$ is the distance from the origin.)
20. Write the system of linear equations as a single matrix equation $AX=B$ (use variable order $x, y$): $\begin{cases} 3x+2y=8\\ x-5y=-3 \end{cases}$

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