Practice Test 50 - Algebra 1

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

Use factoring to analyze $f(x)=x^2-9$ : find the zeros and the axis of symmetry.
Add the rational expressions and simplify (the sum is a rational expression, showing closure under addition): $\frac{2}{x}+\frac{3}{x+1}$
Simplify: $\left(2x^2 + 3x - 1\right) - \left(x^2 - 5x + 4\right)$
What is the explicit formula for the arithmetic sequence with $a_1 = 7$ and common difference $d = -2$ ?
A bacteria population doubles every 3 hours. If the initial population is represented by $P_0$ , which expression is equivalent to $2^{t/3}P_0$ and reveals the approximate growth factor per hour?
What is the minimum value of the function $f(x)=x^2-6x+11$ ? (Find it by completing the square.)
The sequence $(a_n)$ is defined recursively for $n=1,2,3,\dots$ by $a_1=5$ and $a_{n+1}=a_n+4$ . What is $a_6$ ?
What are the terms in $10-2r^2+r$ ?
A gym wants to describe how busy it is throughout a typical day. Which definition is better for modeling "number of visitors," and why?
Use proof by contradiction to justify the claim: "If $r$ is rational and $i$ is irrational, then $r+i$ is irrational." Which option correctly follows this structure?
Construct a viable argument showing that if $7x+1=29$ has a solution, it must be $x=4$ . Which sequence of justifications correctly supports the steps $7x+1=29 \rightarrow 7x=28 \rightarrow x=4$ ?
At what $x$ -value does $f(x)=-(x^2-10x+4)$ reach its maximum? (Use completing the square.)
A profit model is $P = R - C$ , where $P$ is profit, $R$ is revenue, and $C$ is cost. Solve for revenue $R$ in terms of $P$ and $C$ (treat variables like numbers and use the same steps as a numeric equation).
A school club is buying notebooks and pens for a fundraiser. Let $x$ = number of notebooks and $y$ = number of pens. Notebooks cost $3 each and pens cost $2 each. The club has at most $60 to spend, and they need at least 10 total items. Also, $x$ and $y$ must be non-negative whole numbers. Which system of constraints models this situation?
While solving $\frac{2x + 6}{3} = 8$ , a student multiplies both sides by 3 to get $2x + 6 = 24$ . Which statement best explains why this step is valid?
A rectangular garden has area $A(x)=x^2+5x+6$ (in square units). The area is zero when $A(x)=0$ . What are the solutions to $x^2+5x+6=0$ ?
For the quadratic function $y=(x-4)^2$ , what is the vertex of the parabola on the coordinate plane?
Verify a computation using the geometric series formula: Evaluate $S_5=3+6+12+24+48$ by using $S_n=\frac{a(1-r^n)}{1-r}$ with $a=3$ , $r=2$ , $n=5$ .
A bakery is making muffins and cookies for a fundraiser. Let $x$ be the number of dozen muffins and $y$ be the number of dozen cookies. Constraints: - Each dozen muffins uses 2 cups of flour; each dozen cookies uses 1 cup of flour. - The bakery has at most 30 cups of flour: $2x+y\le 30$ . - They need at least 18 dozen items total: $x+y\ge 18$ . - They can make at most 10 dozen muffins: $x\le 10$ . - $x\ge 0$ and $y\ge 0$ . Is $(x,y)=(9,9)$ a viable solution?
An investment is modeled by $A(t)=1500(1.02)^{12t}$ , where $t$ is in years. The base $1.02$ is applied each month. What is the monthly interest rate (as a percent)?

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