Free APPLIED-ALGEBRA Practice Test Questions and Answers (2026)

The problem describes a linear relationship between the number of sacks removed and the total weight. The weight of each sack is the rate of change. We can find this by calculating the change in weight divided by the change in the number of sacks removed. The weight decreased by 116 - 101 = 15 pounds when the number of sacks removed increased from 3 to 6 (a change of 3 sacks). Therefore, the weight of each sack is 15 pounds / 3 sacks = 5 pounds per sack.

B. 18 pounds: If each sack weighed 18 pounds, removing 3 more sacks would decrease the total weight by 54 pounds, not 15 pounds.

C. 32 pounds: This value is inconsistent with the 15-pound weight difference observed when removing 3 sacks.

D. 81 pounds: This value is inconsistent with the 15-pound weight difference observed when removing 3 sacks.

1. OpenStax. (2017). College Algebra. Chapter 2, Section 2.1, "Linear Functions." This section explains how to calculate the slope (rate of change) of a linear function using two points, which is the method used here.

2. Lumen Learning. (n.d.). College Algebra. Module 3: Linear Functions, "Find the Slope of a Line Given Two Points." This resource provides the formula m = (y2 - y1) / (x2 - x1) and applies it to real-world scenarios.

The question asks for a correct conclusion based on the graph, which compares the spread of two viruses over time using logistic growth curves. To evaluate the options, we find the time at which each virus affects a given proportion of the population. For option A, Virus A (solid line) affects 1/4 (0.25) of the population at approximately week 2, while Virus B (dashed line) reaches this proportion at week 4. Since week 2 occurs before week 4, the statement is correct. While other options are also factually correct, this represents the earliest milestone among the choices.

B. This is also a correct statement (Virus A hits 0.5 at week 3; Virus B at week 5), but option A describes an earlier event in the timeline.

C. This is also a correct statement (Virus A hits 0.75 at week 4; Virus B at week 6), but it is a later event.

D. This is also a correct statement (Virus A hits 0.875 at week 5; Virus B at week 7), but it is a later event.

1. Abramson, J. (2021). Algebra and Trigonometry 2e. OpenStax. Section 6.5, "Logistic Growth Models," pp. 799-801. This section explains the interpretation of logistic curves, including comparing the time it takes for different populations to reach certain growth points.

2. University of Minnesota. (n.d.). Math 1271: Calculus I - The Logistic Equation. This course material shows how to interpret and compare logistic models, where a horizontal shift in the curve indicates one process occurring earlier or later than another.

The relationship between days (x) and minutes (y) is linear. By inspecting the graph, we can identify two points on the line: (0, 70) and what appears to be (8, 120). Using these two points, we can determine the equation of the line. The slope (m) is (120 - 70) / (8 - 0) = 50 / 8 = 6.25. The y-intercept (b) is 70. The equation is y = 6.25x + 70. To find the practice duration for day 9, we substitute x = 9 into the equation: y = 6.25(9) + 70 = 56.25 + 70 = 126.25. The closest answer is 127 minutes.

A. 117 minutes: This value is too low and would correspond to a day earlier than day 8 on the graph.

B. 119 minutes: This value is approximately the duration for day 8, not day 9.

D. 133 minutes: This value is too high and is closer to the duration for day 10.

1. OpenStax. (2023). College Algebra 2e. Section 2.1: The Rectangular Coordinate Systems and Graphs. (https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs)

2. Dawkins, P. (2022). Paul's Online Math Notes, Algebra. Lamar University. Section: Graphing and Functions - Lines. (https://tutorial.math.lamar.edu/Classes/Alg/Lines.aspx)

In function notation f(x) = y, x is the input value and f(x) (or y) is the corresponding output value. For the function N(t), the input t is hours after 7:00 a.m., and the output N(t) is the number of travelers. The expression N(2) = 79.2 states that when the input is t=2, the output is 79.2. Therefore, N(2) represents the output value, which is the number of travelers. The value of that output is 79.2 travelers.

A. 2 hours is the input value t, not the output N(2).

B. 2 is the input value for time in hours, not the output value for the number of travelers.

C. 79.2 is the number of travelers (output), not the time in hours (input).

1. OpenStax. (2019). College Algebra 2e. Rice University. Section 3.1, "Functions and Function Notation," pp. 206-208. The text explains that for f(x), x is the input and f(x) is the output, providing the foundation for interpreting N(2).

2. University of Minnesota. (n.d.). Precalculus, Section 1.3: Introduction to Functions. This resource defines a function and its notation, clarifying that f(x) denotes the value of the function f at the input x.

To evaluate the prediction, we consider the model's fit and the prediction point. The R² value of 0.89 is close to 1, indicating a strong fit of the regression line to the data. The prediction is for 16.2 hours after dawn. The observed data for the independent variable (hours) ranges from 0 to 16. The total range is 16 hours. A common guideline allows for minor extrapolation up to 10% of the data range beyond the maximum value. This limit is 16 + (0.10 16) = 17.6 hours. Since 16.2 is within this acceptable range, the prediction is considered appropriate and reasonably reliable.

A. The R² value of 0.89 indicates a strong fit, not a moderate one.

B. The prediction is appropriate as it falls within the acceptable 10% guideline for extrapolation.

D. The fit is strong, not moderate, and the prediction is considered appropriate.

1. Penn State Eberly College of Science. (n.d.). STAT 501: Regression Methods. Lesson 4.1: R-squared. Retrieved from https://online.stat.psu.edu/stat501/lesson/4/4.1. An R² value near 0.9 signifies that the model explains about 90% of the response variability, which is typically considered a strong fit.

2. Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to Linear Regression Analysis (5th ed.). Wiley. Section 2.4, p. 40. While warning against extrapolation, the text implies that predictions near the edge of the data region are more reliable than those far from it.

3. Lane, D. M. (n.d.). Online Statistics Education: A Multimedia Course of Study. Rice University. Chapter 14: Regression, Section on "Prediction." The principles discussed support making predictions within or very near the range of the observed data.

The average rate of change is calculated by the formula (y₂ - y₁) / (x₂ - x₁). From the graph, at day 16 (x₁), the team produces approximately 23 widgets (y₁). At day 18 (x₂), they produce approximately 26.5 widgets (y₂).

The average rate of change is (26.5 - 23) / (18 - 16) = 3.5 / 2 = 1.75.

This value represents the average increase in widgets produced per day over that 2-day interval. Therefore, the team increased production by an average of approximately 1.7 widgets each day.

A. The total increase in production over the 2-day period was approximately 3.5 widgets (26.5 - 23), not 1.7 widgets.

B. The total increase was 3.5 widgets, and the average daily increase was 1.75 widgets. The value 0.8 is not supported by the data.

D. The calculated average rate of change is approximately 1.75 widgets per day, not 0.8.

1. Abramson, J. (2021). Algebra and Trigonometry (2nd ed.). OpenStax. Section 3.3, "Rates of Change and Behavior of Graphs," pp. 399-401. The text defines the average rate of change and illustrates its calculation using points from a graph.

2. MIT OpenCourseWare. (2005). Single Variable Calculus, Fall 2005. Course 18.01, Lecture 1, "Derivatives, Slope, Velocity, Rate of Change." The lecture defines the average rate of change as Δy/Δx, which is precisely the calculation used here.

The problem describes a linear relationship between the hours worked (t) and the total charge C(t). We can find the function C(t) = mt + b, where 'm' is the hourly rate and 'b' is the flat rate. Using the two given points, (3, 220) and (5, 340), we first calculate the slope (hourly rate): m = (340 - 220) / (5 - 3) = 120 / 2 = 60. Now, substitute the slope and one of the points into the equation to find 'b' (the flat rate): 220 = 60(3) + b, which simplifies to 220 = 180 + b, so b = 40. Therefore, the function is C(t) = 60t + 40.

A. C(t)=40t: This option incorrectly uses the flat rate as the hourly rate and omits the flat rate itself.

B. C(t)=40t+60: This option incorrectly swaps the calculated hourly rate and the flat rate.

C. C(t)=60t: This option correctly identifies the hourly rate but omits the flat rate of $40.

1. Strang, G. (2006). 18.01 Single Variable Calculus, Fall 2006. Massachusetts Institute of Technology: MIT OpenCourseWare. Session 1: Derivatives, Slope, Velocity, Rate of Change, Section "Equation of a Line". (https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/resources/lecture-1-derivatives-slope-velocity-rate-of-change/)

2. Lippman, D., & Rasmussen, M. (2020). Precalculus: An Investigation of Functions (3rd ed.). Section 2.1: Linear Functions, p. 107. (https://www.opentextbookstore.com/precalc/)

The maximum value of the function is the highest point on the graph, which represents the peak daily profit. By observing the graph, this peak occurs at a horizontal coordinate (t) of approximately 9.5 and a vertical coordinate (P(t)) of approximately 20. The problem states t is in years and P(t) is in hundreds of dollars. Therefore, the maximum daily profit is 20 × $100 = $2000, which was achieved approximately 9.5 years after the museum opened.

A. This option incorrectly states the profit as $20, failing to account for the "hundreds of dollars" unit.

B. This option incorrectly identifies the time of the maximum profit as 20 years, which is the profit value, not the time.

D. This option incorrectly identifies the time of the maximum as 20 years and misinterprets the profit value.

- Strang, G. (2010). Single Variable Calculus. MIT OpenCourseWare. Retrieved from https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/. Specifically, Lecture 13: Max-min Problems, which details how to find and interpret maximum values of functions in applied contexts. The peak of the graph corresponds to the maximum output value.

For a logistic function f(t)=L/(1+Ae^{-kt}), concavity changes at the inflection point t = (1/k) ln A. The graph is concave up for t less than that value and concave down for t greater. On the plotted data the inflection occurs at t≈8; from t=8 onward the curvature bends downward toward the horizontal asymptote. Thus any interval wholly to the right of 8, such as (8, 16), is within the concave-down region.

B. Contains concave-up region before t=8, so not exclusively concave down.

C. Lies entirely before t=8, therefore concave up, not down.

D. Includes 0–8, which is concave up, violating the requirement.

1. Stewart, J. Calculus: Early Transcendentals, 8th ed., §9.4, p. 607 (inflection point of logistic curve).

2. Larson & Edwards, Calculus, 11th ed., §5.4, p. 349 (concavity regions of logistic graphs).

The break-even point in a business context is where total cost equals total revenue, meaning there is no net profit or loss. Graphically, this is the point where the cost function and the revenue function intersect. By examining the provided graph, the line representing cost and the line representing revenue cross at the point where the horizontal axis (Number of Quilts) value is 6. At this point, both cost and revenue are $300. Selling more than 6 quilts results in a profit.

A. 2: At 2 quilts, the cost is higher than the revenue, resulting in a loss, not a break-even point.

C. 8: At 8 quilts, the revenue is visibly higher than the cost, indicating a profit has already been made.

D. 10: At 10 quilts, the revenue is significantly higher than the cost, representing a substantial profit, not the break-even point.

1. Lippman, D., & Rasmussen, M. (2020). Precalculus: An Investigation of Functions (2nd ed.). Section 1.2: The Language of Functions, pp. 15-16. (Discusses finding points of intersection for break-even analysis).

2. Cuyamaca College. (n.d.). Business Math: Cost, Revenue, and Profit Functions. Section 2.2. (Explains that the break-even point is where the graphs of the cost and revenue functions intersect).