Practice Test 57 - Algebra 2

19 of the 20 questions below are in this test; the other one can't be scored yet. Answer them in any order, then submit to see your score and every explanation.

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1. Solve the quadratic equation by taking square roots: $(x-4)^2=9.$
2. A function $h(x) = -x^2 + 8x - 7$ is defined on the interval $[0, 8]$. To create an invertible function, Maria restricts the domain to $[0, 4]$ and Jose restricts it to $[4, 8]$. Which statement about their restricted functions is correct?
3. How does the graph behave at each zero of $p(x)=(x+1)(x-2)^4?$ Also state the end behavior.
4. Order the following functions by eventual growth rate (from fastest to slowest) as $x\to\infty$: - Linear: $f(x)=20x$ - Quadratic: $g(x)=x^2$ - Polynomial (degree 5): $h(x)=x^5$ - Exponential: $p(x)=1.3^x$​
5. Simplify the expression $\left(2^3\right)^t$ using the power-of-a-power property.​​
6. Verify whether the point $(2,3)$ is a solution to the linear-quadratic system (a solution must satisfy both equations). $\begin{cases} y = x^2 - 1 \\ y = 3x - 3 \end{cases}$ Which statement is correct?
7. Rewrite the logarithmic equation $\log_5(125)=3$ in exponential form using $b^y=x \Leftrightarrow \log_b(x)=y$.​
8. Over $\mathbb{C}$, you can factor expressions using standard identities even when no real factorization exists. Factor $x^4+4$ completely over $\mathbb{C}$.
9. Write the system of equations as a single matrix equation $AX=B$ (use variable order $x, y$): $$\begin{cases} 3x+2y=8\\ x-5y=-3 \end{cases}$$
10. Multiply and simplify using $i^2=-1$: $(3+2i)(1-4i).$
11. Which statement best matches the "Fundamental Theorem for quadratics" idea: a degree-2 polynomial $ax^2+bx+c$ has exactly 2 complex zeros counting multiplicity, with the discriminant $b^2-4ac$ determining whether they are 2 distinct real, 1 repeated real, or 2 complex conjugates? (This also previews the Fundamental Theorem of Algebra: degree $n$ has $n$ complex zeros counting multiplicity.)
12. Which choice correctly identifies $A$, $X$, and $B$ for the system (use variable order $x, y$)? $$\begin{cases} 4x-y=10\\ -3x+6y=-12 \end{cases}$$
13. Factor completely over the complex numbers: $x^4+4$. (Hint: use $a^2+b^2=(a+bi)(a-bi)$ with $a=x^2$ and $b=2$.)
14. A radioactive sample's mass is recorded every day: Day: 0, 1, 2, 3 Mass (g): 50, 46, 42, 38 Does this represent constant percent decay? (Hint: constant percent change means a constant ratio, not a constant difference.)
15. A fireworks shell's height above the ground (in meters) $t$ seconds after launch is modeled by $h(t)=-5(t-2)^2+20$, for $0\le t\le 5$. Identify the maximum height of the shell and when it occurs.
16. Use the table to approximate the solution to $f(x)=g(x)$, where $f(x)=x^3$ and $g(x)=2x+5$. Choose the best estimate for the solution to the nearest tenth.
17. The function $f(x)=7x-3$ represents a relationship between $x$ and $f(x)$. Is the rate of change constant? If yes, what is it?
18. A polynomial function $p(x)$ has the property that $p(x) = x^2(x^2 - 9)(2x - 6)$. When analyzing this function's complete behavior, which statement identifies the most significant graphical feature that distinguishes it from simpler polynomial functions?
19. Factor and graph $p(x)=x^3-4x$ showing all zeros and the correct end behavior.
20. 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 $t$?

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