Practice Test 24 - Algebra 2

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

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1. Which structural form of $27a^3 - 64b^6$ most clearly shows how to factor it completely?
2. The function $f(x)=(x-3)^2+1$ (domain: all real numbers) fails the horizontal line test. What domain restriction makes $f$ invertible?​
3. Add the polynomials (combine like terms). This illustrates the closure property because the sum of two polynomials is also a polynomial: $(2x^4-3x^3+x^2-5)+(x^4+2x^3-4x^2+x+3).$​
4. In the expression $\frac{6x^3 - 9x^2 + 12x}{3x}$, after simplification, how many terms have a coefficient that is a prime number?
5. The function $f(x) = 3 \cdot 2^{x-1} - 4$ is graphed on a coordinate plane. Which statement best describes the key features of this exponential function?
6. The brightness of a star as observed from Earth follows $B(d) = B_0 \cdot 2^{-d/5}$, where $d$ is distance in parsecs. Which transformation shows how the brightness changes when the distance is measured in light-years (1 parsec ≈ 3.26 light-years)?
7. Function $f$ is given by $f(x)=-(x+1)(x-5)$. Function $g$ is described verbally as: "A quadratic function with x-intercepts at $x=-3$ and $x=2$." Which function has the larger greater x-intercept (the larger zero)?
8. Use the Rational Zeros Theorem to help factor the polynomial completely over the integers: $P(x)=x^3-7x-6.$​
9. For the rational function $f(x)=\frac{2x^2-8}{x^2+1},$ what are the zeros and the horizontal asymptote? (Then you could sketch using these features.)
10. Rewrite $\dfrac{(1.10)^{5t}}{(1.10)^{2t}}$ as a single exponential expression using the quotient of powers property.
11. Factor and find the zero(s) of $h(x)=x^2-6x+9$. Then state the x-intercept(s) of the graph.
12. A complex number $z$ satisfies $|z| = 3$ and $\text{arg}(z) = 60°$. When $z$ is divided by $1 + i\sqrt{3}$, what is the modulus of the resulting quotient?
13. A city models the height of tide (in meters) by $T(t)=2\cos\left(\frac{\pi}{6}t\right)+5$, where $t$ is time in hours. What is the period of $T$?
14. Which restriction makes $f(x)=(x+2)^2$ pass the horizontal line test (so it has an inverse)?
15. If $z_1 = 1 + 3i$ and $z_2 = 4 - 2i$, what is $\overline{z_1 + z_2} - \overline{z_1} - \overline{z_2}$?
16. In the expression $4y(3y + 1) - 2(y^2 - 6)$, what is the coefficient of the $y^2$ term when the expression is simplified?
17. A phone's value depreciates over time according to $V(t)=900(0.85)^t$, where $V$ is in dollars and $t$ is time in years. In the function $V(t)=900(0.85)^t$, interpret the parameter 900.
18. Describe the graph's behavior based on factored form for $p(x)=(x+2)(x-1)^2(x-3).$ Which statement is correct about zeros, multiplicities, crossing/touching, and end behavior?
19. Compute the modulus-squared connection for $z=6-2i$ by finding $z\overline{z}$. (Recall $z\overline{z}=a^2+b^2$ for $z=a+bi$.)
20. Convert between exponential and logarithmic forms: $\log_2(8)=3$. Which exponential equation is equivalent?

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