Practice Test 87 - Algebra 2

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

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1. Determine whether the system is consistent. If it has a solution, find it; otherwise, state no solution. $\begin{cases} 3x + 2y = 10 \\ 6x + 4y = 25 \end{cases}$
2. Verify algebraically that for the exponential function $g(x)=4\cdot(1.5)^x$, the growth factor over equal intervals of length $h=2$ is constant by simplifying $\dfrac{g(x+2)}{g(x)}$.
3. Two different relationships are shown in tables with equal $x$-intervals. Table 1: $x$: 0, 1, 2, 3 $y$: 4, 9, 14, 19 Table 2: $x$: 0, 1, 2, 3 $y$: 2, 4, 8, 16 Which table shows a constant rate of change (linear)?
4. The function $g(x) = x^4 - 8x^2 + 16$ can be written as $g(x) = (x^2 - 4)^2$. To make this function invertible by domain restriction while ensuring the resulting function passes through the point $(\sqrt{3}, 1)$, which restriction is most appropriate?
5. Order these functions by eventual growth rate (fastest to slowest) as $x\to\infty$: $f(x)=0.5x^4$, $g(x)=100x^2$, $h(x)=7x$, and $p(x)=1.1^x$.
6. For a finite geometric series with first term $a$ and common ratio $r$ (where $r\ne 1$), a student writes: $S_n=a+ar+ar^2+\cdots+ar^{n-1}$ $rS_n=ar+ar^2+\cdots+ar^n$ $S_n-rS_n=a-ar^n$ Which is the correct next step to solve for $S_n$?​​​
7. The function $f(x) = x^2$ is transformed to create $g(x) = -2f(x-3) + 1$. If the vertex of $f(x)$ is at $(0,0)$, what are the coordinates of the vertex of $g(x)$?
8. Let the parent function be $f(x)=\dfrac{1}{x}$. What is the effect of replacing $f(x)$ with $g(x)=f(-x)$?​
9. Let $f(x)=x^3$ be the parent function. What transformation changes $f(x)$ to $g(x)=-f(x)$?
10. Let $z_1 = 5 + 3i$ and $z_2 = 2 - 7i$. What is $z_1 + z_2$ in standard form $a+bi$?
11. Determine whether the function $p(x)=x^3-x$ is even, odd, or neither.
12. Predict the solution type using the discriminant, then solve in $a\pm bi$ form: $x^2 - 6x + 10 = 0$.​
13. Two savings plans are described below for the account balance after $t$ months. Plan 1: $A(t)=500+40t$ Plan 2: $B(t)=500(1.08)^t$ Which plan shows a constant percent rate of change per month?
14. In the expansion of $(2a - 3b)^5$, which statement about the signs of the terms is correct?
15. The table compares four functions: a linear function $f(x)=5x$, a quadratic function $g(x)=x^2$, a polynomial function $h(x)=x^4$, and an exponential function $p(x)=2^x$. For sufficiently large $x$ (long-term behavior), which ordering of growth rates is correct?

    $x$	$f(x)=5x$	$g(x)=x^2$	$h(x)=x^4$	$p(x)=2^x$
    2	10	4	16	4
    4	20	16	256	16
    8	40	64	4096	256
    16	80	256	65536	65536
    20	100	400	160000	1048576

16. On the complex plane (real axis horizontal, imaginary axis vertical), the complex number $z = -3 + 4i$ is represented by the point $(a,b)$ where $z=a+bi$. What is the modulus $|z|$ (the distance from the origin), and which point represents $z$?​
17. A quantity is modeled by $y=600(0.97)^t$, where $t$ is measured in days. Which statement correctly describes the situation?
18. A ball is thrown upward from a platform. Its height above the ground after $t$ seconds is modeled by $h(t)=-16t^2+48t+10$, where $h$ is in feet. Graph the function for $0\le t\le 4$ with appropriate labels and a reasonable scale. Which choice shows the best axes labels and scale for this graph?
19. A rectangular garden has perimeter $40$ ft. Let $L$ be the length (ft) and $W$ be the width (ft). Write an equation relating $L$ and $W$, then choose appropriate axes labels and a reasonable scale to graph all solutions with $0\le L\le 20$.
20. Solve for $t$ and express the solution as a logarithm. Then use a calculator to approximate: $12\cdot 2^{5t}=150$.

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