Mat 121 College Algebra Written Assignment 32 Points Each

Mat 121 College Algebrawritten Assignment 32 Points Each Except For 5

Mat 121 College Algebra written assignment with various questions focusing on functions, graphs, algebraic manipulation, real-world applications, and intercepts. The tasks include determining whether relationships are functions, writing functions in different forms, evaluating functions at specific points, analyzing graphs, and solving algebraic and real-world problems using functions and transformations. Some exercises require finding domains, ranges, inverses, and intercepts, as well as interpreting functions in context. The assignment also covers analyzing slope, linear equations, and transformations of functions, along with applications such as projectile height, garden compost, and balloon radius.

Paper For Above instruction

The assignment seeks to assess a student's understanding of fundamental concepts in algebra and functions, including the ability to analyze relationships for their functional status, manipulate algebraic expressions, interpret graphs, and apply functions to real-world situations. These skills are essential in developing mathematical literacy necessary for advanced studies in mathematics, engineering, physics, and related fields.

Particularly, students are asked to determine whether given relationships qualify as functions, which involves understanding the definition of a function and the criteria for multiple outputs for a single input. For instance, analyzing relationships like x + y² = 5 and 7n - 5m = 4 involves solving for one variable in terms of the other and writing it explicitly as a function, such as n = k(m), and evaluating the function at specific points.

Graphical problems require students to interpret and analyze the behavior of functions visually, such as estimating where functions are increasing or decreasing, and understanding the implications of the graph's shape on the function's domain and range. This includes work with projectile height functions, where the physical interpretation of the domain relates to the time the projectile is in the air, and the range relates to the maximum height achieved.

Advanced topics like inverse functions are also covered, where students are asked to find the inverse of functions, understand the restrictions necessary for such inverses to exist, and interpret these inverses both algebraically and graphically. These skills are critical in understanding one-to-one functions and their applications.

Transformation of functions is another emphasis of the assignment, requiring students to describe shifts, reflections, stretches, or compressions of basic functions such as quadratics and linear functions. Such transformations are fundamental in understanding how functions behave under various manipulations and are widely applicable in real-world modeling.

Finally, the assignment incorporates real-world applications, such as modeling the amount of compost needed for a garden, the radius of a balloon based on volume, and the tip earnings of a waitress based on the number of customers served. These applications demonstrate the practical relevance of algebraic concepts and reinforce the importance of functions in describing various phenomena.

In-depth analysis and solutions

In analyzing the given relationships, the first step is to determine whether they can be written explicitly with y or the relevant dependent variable as a function of the independent variable. For example, the relationship x + y² = 5 can be solved for y as y = ±√(5 - x). Since for each x in the domain where 5 - x ≥ 0, there are two possible y-values (positive and negative square roots), this relationship is not a function because it assigns two outputs to a single input.

Similarly, relationships like 7n - 5m = 4 can be solved for n in terms of m as n = (7m + 4) / 5, which is a valid function of m. Evaluating this at m = 5 leads to n = (7*5 + 4) / 5 = (35 + 4)/5 = 39/5 = 7.8, illustrating how to compute specific outputs once the function is established.

Graphical analysis involves interpreting the shape and features of graphs. If a graph shows a function, then for each x-value, there is exactly one y-value. For example, evaluating f(4) on a graph involves locating x=4 and reading off the y-value from the graph. Estimating where the function is increasing or decreasing aids in understanding its behavior, such as where a projectile reaches its maximum height, which corresponds to the vertex of a parabola in the height function h(t)=-9.8t²+19.6t.

In assessing whether a relationship like {(0, 5), (-5, 8), (0, -8)} represents a function, the key is to check for multiple y-values for a single x, which is apparent in the pair (0, 5) and (0, -8). Since the same x-value corresponds to different y-values, this is not a function.

Functions like f(x)=4x+2 and g(x)=7-6x when evaluated at specific points, such as f(-6) or g(1), permit direct calculation, reinforcing algebraic manipulation skills. Furthermore, combining functions through operations—addition, subtraction, multiplication, or composition—requires understanding their domains and the behavior of composite functions like g(f(h(x))). For instance, plugging h(x)=2x into f(x)=x³-3 involves substituting h(x) into f to find a composite expression and simplifying.

In real-world problems, functions translate into models describing physical or economic phenomena. For the garden's compost problem, the relationship C=h(A) specifies how much compost is needed based on the garden area A. Given data points allow deriving the function h(A), e.g., h(2500)=12.5 indicates that for a garden with an area of 2,500 square feet, 12.5 cubic yards of compost are needed.

The projectile height function h(t)=-9.8t²+19.6t models the height of an object / projectile over time. The domain in seconds depends on the physical scenario—here, it is the time interval where the height is non-negative, i.e., from t=0 to the time when height returns to zero after reaching the maximum. Solving the quadratic equation h(t)=0 gives these critical points, which indicate the start and end times of the projectile's flight.

Rate of change calculations, especially average rates over intervals, are central to analyzing how functions behave over ranges. For the moon's object fall described by d(t)=1.6t², the average velocity from t=2 to t=5 is (d(5)-d(2))/(5-2), which corresponds to the change in distance over the change in time, providing insights into the acceleration due to gravity and motion under free fall conditions.

Transformations of functions, such as shifting, stretching, or reflecting, involve altering the basic functions and understanding how these transformations affect their graphs and equations. For example, shifting the graph of g(x)=x by 3 units up and 1 unit left results in a new function g(x)= (x+1)+3, or simplified to g(x)=x+4, which shifts the graph accordingly. Analyzing these transformations helps in understanding the behavior of complex functions built from basic ones.

When considering inverse functions, the critical step is to solve for x in terms of y. For a linear function like f(x)= -2x-13, its inverse f⁻¹(x) can be found by swapping x and y and solving for y, leading to f⁻¹(x)= (-x-13)/2. Restrictions often apply to ensure the original function is one-to-one. For example, restricting the domain of f(x)= x+112-14 to make it invertible involves choosing an interval where the function is strictly increasing or decreasing, thus guaranteeing a unique inverse function.

In applications such as the area of a circle, expressing the radius as a function of area, r(A)=√(A/π), demonstrates the practical use of inverse functions. Calculating r(64) provides numerical values that can be interpreted physically, such as the radius of a circle with a known area.

Overall, this set of exercises embodies a comprehensive review of algebraic and analytic skills, emphasizing the importance of understanding relationships, transformations, graphs, and applications of functions in real-world contexts. These foundational skills are crucial for advanced mathematical reasoning and problem-solving in scientific and engineering domains.

References

• Anton, H., Bivens, I., & Davis, S. (2019). Calculus: Early Transcendentals (11th ed.). Wiley.
• Larson, R., & Edwards, B. H. (2018). Precalculus with Limits: A Graphing Approach (8th ed.). Cengage Learning.
• Stewart, J. (2016). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
• Mitchell, S. (2017). Algebra and Trigonometry. Pearson.
• Sullivan, M., & Miranda, J. (2018). Precalculus. Pearson.
• Swokowski, E., & Cole, J. (2016). Algebra and Trigonometry with Analytic Geometry. Brooks/Cole.
• Roth, H., & others. (2020). Algebra and Functions for Calculus. Wiley.
• Finzer, P. (2012). Graphs and Functions: An Introduction for College Students. Springer.
• Fletcher, D., & others. (2015). Precalculus: Concepts Through Functions. Pearson.
• Lay, D. C. (2016). Linear Algebra and Its Applications (5th ed.). Pearson.