Practice Test 98 - Algebra 2

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

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1. An exponential function satisfies $f(0)=7$ and $f(1)=14$. Write the function in the form $f(x)=a\cdot b^x$.
2. In the compound interest model $A = P(1+r)^n$, $P$ is the initial deposit, $r$ is the interest rate per period, and $n$ is the number of periods. Interpret the expression by chunking it into meaningful parts. Which statement correctly views the formula as a product of two entities and identifies what does not depend on $P$?
3. What is the product of a complex number and its conjugate? Compute $(2 + 5i)(2 - 5i)$ and simplify.
4. Add and write the result in standard form (this illustrates the closure property because the sum of two polynomials is a polynomial): $(2x^4 - 3x^3 + x^2 - 5) + (x^4 + 2x^3 - 4x^2 + x + 3)$
5. A function $T(t)$ models the temperature difference (in °C) from a comfortable baseline in a greenhouse, where $t$ is hours after midnight. The temperature difference repeats every 24 hours. If $T(3)=2$, what is $T(27)$?
6. The intensity of sound is measured in decibels using $dB = 10\log\left(\frac{I}{I_0}\right)$, where $I$ is the sound intensity and $I_0$ is a reference intensity. If one sound is 40 dB and another is 70 dB, how many times more intense is the louder sound?
7. For the function $g(x)=-2x^3+5x$, describe the end behavior of $g(x)$.
8. Consider the matrix equation: $\begin{pmatrix} 2 & -1 & 3 \\ 0 & 4 & -2 \\ 1 & 1 & 0 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \\ 4 \end{pmatrix}$ If this represents a system of linear equations, what is the constant term in the equation that has no $c$ variable?
9. Factor $h(x)=x^2-6x+9$ and use the zero product property to find its zero(s). How many x-intercepts does the graph have?
10. Verify or disprove: $f(x)=\dfrac{1}{x-2}$ and $g(x)=\dfrac{1}{x}+2$ are inverse functions. Compute and simplify $f(g(x))$ and $g(f(x))$ (assume $x$ is in the domain of each composition).​
11. Simplify using conjugates and write your answer in $a+bi$ form: $\frac{3+i}{2-i}.$
12. Identify the amplitude, period, and midline of the trigonometric function $f(x)=-3\sin(2x)+1$.​
13. Use the Rational Zeros Theorem to list all possible rational zeros of $P(x)=4x^3-3x^2-8x+6.$ (List candidates $p/q$ in lowest terms.)
14. Verify a formula application: Which value equals the sum $3+6+12+24$ using $S_n=\dfrac{a(1-r^n)}{1-r}$ with $a=3$, $r=2$, $n=4$?
15. What is the horizontal asymptote of the exponential function $f(x)=2^x+3$?
16. What is $x$ in the equation $7e^{2x}=100$? Express your answer as a logarithm (natural log) and evaluate using technology.
17. A sequence can be viewed as a discrete function whose domain is a subset of the integers. For the Fibonacci sequence defined by $f(1)=f(2)=1$ and $f(n)=f(n-1)+f(n-2)$ for $n\ge 3$ (with domain $n\in\{1,2,3,\dots\}$), find $f(7)$.
18. Expand the expression using the product, quotient, and power properties: $\ln\left(\frac{a^3\sqrt{b}}{c}\right).$
19. A polynomial has factored form $p(x)=3(x+2)^2(x-1)(x-4).$ Sketch $p(x)$ showing x-intercepts and end behavior. Which option correctly lists the intercept behavior and end behavior?
20. Given the explicit formula $a_n=9+3(n-1)$ for $n\ge 1$, write the equivalent recursive definition.

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