Practice Test 48 - Algebra 2

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

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1. Simplify using the inverse property $b^{\log_b(x)}=x$ (for $x>0$): $2^{\log_2(17)}$.
2. A city's population is recorded every 5 years: Year: 0, 5, 10, 15 Population: 50,000; 53,000; 56,000; 59,000 Classify the pattern as exponential growth, exponential decay, or neither. (Hint: compare differences vs ratios.)
3. Identify $A$, $X$, and $B$ for the system written in the order $(x, y, z)$: $\begin{cases} 5x-2y+z=0\\ -y+4z=12\\ 3x+0y-2z=-6 \end{cases}$ Which choice correctly gives the matrix equation $AX=B$?
4. Use synthetic division to test whether $(x+1)=(x-(-1))$ is a factor of $P(x)=x^3+2x^2-x-2.$ Use $c=-1$ with coefficients $1,2,-1,-2$. The remainder (last entry) equals $P(-1)$ by the Remainder Theorem. Which statement is correct?
5. Prove that any linear function $f(x)=mx+b$ grows by equal differences over equal intervals. Specifically, for a fixed interval length $h$, simplify $f(x+h)-f(x)$ and identify what (if anything) depends on $x$.
6. Recognize and factor completely using polynomial identities: $x^4-16$.
7. What are the center and radius of the circle in standard form $(x-7)^2+(y+2)^2=49$?
8. An investment grows by 12% per year, so its value after $t$ years is multiplied by $(1.12)^t$. Rewrite $(1.12)^t$ to show an equivalent form with a monthly growth factor raised to the number of months, and identify the correct expression. (Use the power-of-power property.)
9. Which identity allows you to factor the expression $a^2-b^2$ as a product of two linear factors?​
10. Are the functions $f(x)=\dfrac{x-1}{x+2}$ and $g(x)=\dfrac{1+2x}{1-x}$ inverses? Verify by computing and simplifying both $f(g(x))$ and $g(f(x))$ (assume $x$ is in the domain of each composition).
11. Describe the end behavior of the exponential function $h(x)=5\cdot(0.6)^x$.
12. The function $f(x)=\sqrt{x-4}$ is used to model the length (in meters) of a cable as a function of $x$, where $x$ is a measurement value. Compare the mathematical domain of $f$ to a realistic domain if $x$ represents a physical length measurement (so $x$ cannot be negative). Which choice is correct?
13. A one-to-one function $f$ is given by the table, so $f$ has an inverse. What is $f^{-1}(14)$?

    $x$	$f(x)$
    $1$	$-2$
    $2$	$4$
    $3$	$9$
    $4$	$14$
    $5$	$20$

14. The parent function $f(x)=\sqrt{x}$ has its graph starting at $(0,0)$ and increasing to the right. Which description matches the transformation to $g(x)=f(x)+3$?
15. Graph $f(x)=\frac{x+1}{x-2}$ identifying the zero, the vertical asymptote, and the horizontal asymptote.
16. Rewrite using exponent structure. Express $8^{2x}$ as a power of 2: $8^{2x}$
17. A ride-share fare (in dollars) is modeled by $F(t)=\frac{18}{6-t}+3,$ where $t$ is the time (in hours) since the ride started. The model is only intended for the first 5 hours of the ride. What is an appropriate realistic domain for $F$?
18. Find the sum of the first 8 terms of the geometric series with first term $a=12$ and common ratio $r=\tfrac{1}{2}$. (Use $S_n=\dfrac{a(1-r^n)}{1-r}$ for $r\ne 1$.)​​​
19. A savings account balance follows the function $B(t)=1500(1.03)^t$, where $t$ is in years. What is the constant percent rate of change, and is it growth or decay?
20. Solve the quadratic equation $x^2 + 2x + 10 = 0$ and express the solutions as complex numbers in $a + bi$ form. (The discriminant is negative.)

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