Practice Test 12 - Algebra 1

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

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1. Which form best reveals why $49x^2 - 14x + 1$ and $49x^2 + 14x + 1$ have a special relationship?
2. A line is graphed on a coordinate plane using the equation $y = 2x - 5$. At what point does the graph cross the y-axis?
3. A material's stretch factor is modeled by multiplying a constant calibration factor and an exponential response. Let $f(x)=3$ and $g(x)=2^x$. Find $(f\cdot g)(x)$.​
4. Solve graphically and algebraically: the line $y=2x-4$ intersects the parabola $y=x^2-4$. What are the intersection point(s)? $\begin{cases} y = 2x - 4 \\ y = x^2 - 4 \end{cases}$
5. In physics, Newton's second law is $F = ma$, where $F$ is force, $m$ is mass, and $a$ is acceleration. Solve for $m$ in terms of $F$ and $a$.
6. A student claims that $8^{2/3} = (8^2)^{1/3}$ and calculates this as $(64)^{1/3} = 4$. Another student claims that $8^{2/3} = (8^{1/3})^2$ and calculates this as $(2)^2 = 4$. Which statement best explains why both methods give the same result?
7. The quadratic equation $ax^2 + bx + c = 0$ has discriminant $\Delta = 0$. If this equation is written in the form $(px + q)^2 = 0$ by completing the square, what is the relationship between the solution and the coefficients?
8. Solve the inequality: $-2x + 8 \le 4$
9. Rewrite the rational expression $\frac{x^3+2x^2-1}{x+1}$ in the form $q(x)+\frac{r(x)}{x+1}$ using polynomial long division (or another valid method). Make sure $\deg(r)<\deg(x+1)$.
10. Describe the solution region for the system $\begin{cases} y \ge 0 \\ y \le -x + 3 \end{cases}$
11. How many real solutions does the linear-quadratic system have? (Equivalently: how many intersection points are there between the line and the parabola?) $\begin{cases} y = x^2 + 1 \\ y = x \end{cases}$
12. To determine shading for the inequality $y> -3x+2$, a student tests the point $(0,0)$. What does this test point indicate?
13. A phone plan charges a one-time activation fee plus a monthly fee. The total cost after $m$ months is $T(m)=35+25m$, where $m$ is the number of months since activation. What is an appropriate realistic domain for $T(m)$?
14. Refer to the graph showing three functions over the domain $0 \leq x \leq 6$. Function $f$ appears to be linear, function $g$ appears to be exponential, and function $h$ appears to be quadratic. At $x = 6$, function $h$ has the highest value, followed by $f$, then $g$. What would be the most likely ordering of the function values at $x = 15$?
15. Calculate the sum of the geometric series $2+6+18+54$. (This is a geometric series, meaning a sum; the corresponding sequence is $2,6,18,54$.) Use $a=2$, $r=3$, $n=4$ and $S_n=\frac{a(1-r^n)}{1-r}$.
16. Solve the system: $\begin{cases} x+y=7 \\ x-y=1 \end{cases}$ What is the solution $(x,y)$ that satisfies both equations?​
17. In the expression $6x(x-3)$, what are the factors being multiplied?
18. A car's value is modeled by $V(t)=20000(0.90)^t$, where $t$ is in years. What is the annual percent depreciation rate?
19. A town's population is modeled by $P(t)=12000(1.02)^t$, where $t$ is in years. Which description is correct?
20. The number of active users $U$ (in millions) on a social media platform $t$ years after launch is modeled by $U(t) = 0.5(4)^{t/2}$. The platform's growth team is analyzing user acquisition patterns. What does the expression $\frac{t}{2}$ in the exponent indicate about the platform's user growth cycle compared to a model with just $4^t$?

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