Algebra 1 — Testing-Out Examination

A student who passes this examination has demonstrated mastery of the Common Core Algebra 1 standards (CCSS-M conceptual categories N-Q, A-SSE, A-APR, A-CED, A-REI, F-IF, F-BF, F-LE, S-ID) and is eligible to advance to Geometry.

Student Name

Proctor

Score

Instructions

Time limit: 120 minutes. Part I (no-calculator) about 45 minutes; Part II (calculator-permitted) about 75 minutes.
Show all work. An unsupported numerical answer earns at most half credit. Box or circle final answers.
Express exact values exactly (e.g., $\sqrt{2}, \pi$) unless rounding is requested.
For graphs, label both axes (with units when appropriate) and at least one specific point on each line/curve.

Part I · No Calculator · Problems 1–4 · 35 pts · ~45 min

1. Linear equations, inequalities, and absolute value 8 points

Solve for $x$: $ \displaystyle \frac{2x - 5}{3} - \frac{x + 1}{4} = 2 $. Show every step; clear fractions before isolating the variable.
Solve the compound inequality $ -3 \le 2x - 7 < 9 $. Graph the solution on a number line.
Solve $ |3x - 4| = 11 $. Show both cases.
Solve $ |2x + 1| \le 5 $ and graph the solution on a number line. State the solution as a single compound inequality and as an interval.

2. Systems of linear equations 9 points

Solve by substitution: $ \begin{cases} y = 3x - 4 \\ 2x + y = 11 \end{cases} $.
Solve by elimination: $ \begin{cases} 3x + 2y = 12 \\ 5x - 2y = 4 \end{cases} $.
Determine without solving: how many solutions does $ \begin{cases} 6x - 4y = 10 \\ 9x - 6y = 15 \end{cases} $ have? Justify by comparing slopes and intercepts.
Write a system of equations to model the following situation, then solve it. Two pencils and three notebooks cost $13.40. Five pencils and one notebook cost $8.80. Find the price of one pencil and one notebook. Define your variables clearly.

3. Polynomial & quadratic algebra 10 points

Multiply and simplify: $(2x - 5)(3x + 4)$. Show the distribution carefully (FOIL or area model).
Factor each completely:
(i) $x^2 - 49$ (ii) $x^2 - 7x + 12$ (iii) $2x^2 + 5x - 12$ (iv) $3x^2 - 27$
Solve $x^2 - 5x - 14 = 0$ by factoring. Show each step.
Convert $f(x) = x^2 - 6x + 1$ into vertex form by completing the square. Show every step. Then state the vertex of the parabola $y = f(x)$.

4. Function notation and key features 8 points

Let $f(x) = 3x - 7$. Compute $f(2)$, $f(-1)$, and find the value of $x$ for which $f(x) = 11$. Show your work for each.
A piecewise function is defined by \[ g(x) = \begin{cases} 2x + 1, & x < 0 \\ x^2, & 0 \le x \le 3 \\ 3x - 1, & x > 3 \end{cases} \] Compute $g(-2)$, $g(0)$, $g(3)$, and $g(5)$. State the domain of $g$.
Sketch the graph of $g(x)$ on a single coordinate plane. Label any open or closed dots at the boundary points $x = 0$ and $x = 3$.

Part II · Calculator Permitted · Problems 5–10 · 65 pts · ~75 min

5. Linear modeling — taxi pricing 10 points

City Cab charges a $3 base fare plus $2.50 per mile. A competing service, Lyft Express, charges a $5 base fare plus $2.00 per mile. Let $C(m)$ and $L(m)$ denote the cost (in dollars) of an $m$-mile trip with each service.

Write linear functions $C(m)$ and $L(m)$. Identify which feature of each function represents the per-mile cost and which represents the base fare.
Compute $C(8)$ and $L(8)$. Which service is cheaper for an 8-mile trip?
A passenger paid $25.50 using City Cab. How many miles did they travel? Show the equation you set up.
For what mile total $m$ do the two services cost the same? Solve algebraically.
Sketch $C(m)$ and $L(m)$ on the same axes for $0 \le m \le 20$. Label intercepts, intersection point, and indicate which service is cheaper on either side of the intersection.

6. Quadratic application — projectile 10 points

A football is kicked straight up from ground level with an initial vertical velocity of 60 ft/s. Its height $h$ (in feet) above the ground at time $t$ (in seconds) is modeled by \[ h(t) = -16 t^2 + 60 t. \]

Compute the height at $t = 1, 2,$ and $3$. Present in a small table.
Find the time $t$ at which the ball reaches its maximum height, and find that height. Use the vertex formula $t = -b/(2a)$ and show your work.
For what value of $t > 0$ does the ball return to the ground? Solve $h(t) = 0$ by factoring.
For what time(s) is the ball exactly 50 ft above the ground? Use the quadratic formula and round each solution to the nearest hundredth of a second.

7. Exponential functions — bacterial growth 10 points

A bacterial culture starts with 500 cells and doubles every 3 hours.

Write a function $P(t)$ giving the population after $t$ hours. Use base-2 form: $P(t) = 500 \cdot 2^{t/3}$. Justify each part of the formula in one sentence.
How many cells are present after 12 hours? After 24 hours? Show your work.
Build a table of values of $P(t)$ at $t = 0, 3, 6, 9, 12, 15, 18, 21$ hours. Use the table to estimate the smallest whole hour count at which the population exceeds 50{,}000.
Suppose a different culture grows linearly at 200 cells per hour starting from 500 cells. Write the linear model $L(t)$. Then find (graphically or by table) the approximate time at which the two cultures have the same population. Sketch both functions on the same axes for $0 \le t \le 24$, and identify on the sketch the time of intersection.

8. Systems of linear equations — application 10 points

A school is selling tickets to a fundraiser. Student tickets cost $5; adult tickets cost $8. They sold 100 tickets total and raised $620 in total revenue. Let $s$ be the number of student tickets sold and $a$ be the number of adult tickets sold.

Write a system of two linear equations in $s$ and $a$ describing the situation.
Solve the system using substitution. Show every step.
Solve the same system using elimination. Confirm you get the same answer.
Verify your solution by substituting both values back into both equations.

9. Statistics — univariate & bivariate 12 points

Part A · Univariate. A school board member is reviewing the end-of-course exam results for a single Algebra 1 class to validate the teacher's claim that "the typical score is in the low 80s." The 12 students earned the following scores: \[ 65, \ 72, \ 75, \ 78, \ 80, \ 80, \ 82, \ 85, \ 88, \ 90, \ 92, \ 95. \]

Compute the mean and the median. Show your work.
Give the five-number summary (min, Q1, median, Q3, max).
Sketch a box plot above a number line. Label all five summary values.
Is the distribution roughly symmetric, skewed left, or skewed right? Justify in one sentence using either the relationship between mean and median or the spacing of the five-number summary.

Part B · Bivariate. Eight students recorded their study hours $x$ and their test score $y$:

Hours $x$	2	3	4	5	6	7	8	9
Score $y$	70	72	78	80	85	88	92	94

A line of best fit for these data is given by $ \hat y = 3.40x + 64.46 $. Interpret the slope in context.
Use the given line to predict the test score for a student who studies 5.5 hours. Round to the nearest whole point.
Describe the direction (positive or negative) and the strength (strong or weak) of the relationship by inspecting the scatterplot. Justify in one sentence.

10. Radicals, rational exponents, and a function in context 8 points

The period $T$ (in seconds) of a simple pendulum of length $L$ (in meters) is approximately \[ T = 2\pi \sqrt{\frac{L}{9.8}}. \]

Find the period when $L = 0.25$ m. Round to two decimals.
Solve the formula for $L$ in terms of $T$. Show every algebraic step.
If a clockmaker wants the pendulum to have a period of exactly 2 seconds, what length $L$ is needed? Round to three decimals.
Express $\sqrt{L / 9.8}$ using rational exponents — i.e., rewrite the formula for $T$ without using a radical sign. Then simplify $ (T/(2\pi))^2 \cdot 9.8 $ using exponent rules to confirm your answer in (b).