Practice Test 3 - Algebra 2

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

The quadratic $4x^2 - 4x - 15$ factors into the form $(2x + a)(2x + b)$ where $a$ and $b$ are integers. What are the zeros revealed by this factorization?
The equation $x^2 + 4x + 13 = 0$ has complex solutions. When written in the form $a \pm bi$ , what are the values of $a$ and $b$ ?
Verify the inverse relationship between the temperature conversion formulas $C(F)=\dfrac{5}{9}(F-32)$ and $F(C)=\dfrac{9}{5}C+32$ by composition. Show that $C(F(C))=C$ and $F(C(F))=F$ .
The equation $x^2 - 8x + k = 0$ is solved by completing the square. If the resulting form is $(x - 4)^2 = 9$ , what is the value of $k$ , and what method would be most efficient for solving the original equation?
A bacteria culture grows according to $N(t) = 500\cdot 2^t$ , where $t$ is time in hours and $N$ is the number of bacteria. In the function $N(t) = 500\cdot 2^t$ , what does the parameter $500$ represent?
Use the inverse relationship $b^y=x \Leftrightarrow \log_b(x)=y$ to evaluate $\log_2(32)$ .
Which statement is always true for a linear function $f(x)=mx+b$ when using any fixed equal interval $h$ ?
Simplify and write the result in standard form. The expression uses only addition, subtraction, and multiplication, so by closure the result is a polynomial: $\left(x+2\right)^2-\left(x-1\right)\left(x+3\right).$
A water tank is being filled at a constant rate. The amount of water is modeled by $W = 15t + 120$ , where $t$ is time in minutes and $W$ is in liters. What does it mean that the y-intercept is $120$ in this context?
Let the parent function be $f(x)=x^3$ . What is the effect on the graph of replacing $f(x)$ with $g(x)=f(x+2)$ ?
Polynomial identities that are true for real numbers also hold for complex numbers because addition and multiplication work the same way in $\mathbb{C}$ . Use the identity $(a+b)^2=a^2+2ab+b^2$ to verify it for the complex values $a=1+i$ and $b=2-i$ . Which option shows a correct verification (both sides match)?
Function $f$ is given by $f(x)=(x-2)^2-9$ . Function $g$ is shown on the coordinate plane as a parabola with vertex at $(0,-4)$ opening upward. Which function has the smaller minimum value?
Factor the quadratic $x^2+25$ over the complex numbers. Recall: every quadratic $ax^2+bx+c=0$ has exactly 2 complex solutions counting multiplicity (either 2 real, 1 repeated real counted twice, or 2 complex conjugates). Which option gives the correct factorization over $\mathbb{C}$ and the zeros?
Use polynomial long division (or inspection) to rewrite the rational expression $\dfrac{2x^3+3x^2-5x+7}{x-2}$ in the form $q(x)+\dfrac{r(x)}{x-2}$ , where $\deg(r)<\deg(x-2)$ .
What is the horizontal asymptote of the exponential function $f(x)=3\cdot 2^x-4$ ? (Think of the parent function $y=2^x$ and how vertical shifts affect the asymptote.)
Add and simplify to lowest terms. This reinforces that rational expressions are closed under addition (the result is still a rational expression). Add: $\frac{x}{x-1}+\frac{2}{x+1}$ (Assume $x\neq \pm 1$ .)
Compute the sum using the finite geometric series formula: $2+2(1.1)+2(1.1)^2+\cdots+2(1.1)^9.$
Prove the identity $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ by expanding the left-hand side. Which expression is the correct expansion of $(a-b)^3$ ?
A complex number is $z=6- i$ . Which expression equals $|z|^2$ (the modulus squared) using conjugates?
Recognize the pattern and factor (difference of cubes): $x^3-27$

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