MATH 1101 – Content Standards
Title: Introduction to Mathematical Modeling
Course Description: Prerequisite: High School algebra II or equivalent. This course is NOT an appropriate prerequisite for precalculus or calculus courses.
Mathematical modeling using graphical, numerical, symbolic, and verbal techniques to describe and explore real-world data and phenomena. Emphasis is on the use of elementary functions to investigate and analyze applied problems and questions, on the use of appropriate supporting technology, and on the effective communication of quantitative concepts and results.
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1. Algebra. Students will demonstrate the ability to:
a. Graph points.
b. Graph linear, piecewise linear, exponential, logarithmic, and quadratic equations and functions. and identify horizontal asymptotes.
c. Determine the equation of a line given two points or one point and the slope.
d. Determine the absolute value of a quantity.
e. Solve and estimate solutions to linear, quadratic, exponential, and logarithmic equations, including use of the properties of exponents and common and natural logarithms.
f. Solve linear systems of two equations by substitution and elimination, including systems that have a unique solution, no solution, or many solutions.
g. Simplify expressions using the laws of exponents and logarithms.
h. Calculate average rate of change of any function.
i. Perform arithmetic calculations to answer questions regarding two-variable data presented in tabular, graphical, or equation form.
j. Express and compare very large and very small numbers using scientific notation and orders of magnitude.
k. Employ the relationship y = bx ⇔log b y = x to solve exponential and logarithmic equations.
l. Factor quadratic expressions.
m. Complete the square of quadratic expressions.
n. Express the square root of negative numbers in terms of the imaginary unit, i.
o. Given conversion factors, convert units of measure.
p. Use the quadratic formula to solve quadratic equations.
2. Functions. Students will demonstrate:
a. An understanding of the definitions of function, domain, range, independent and dependent variables, and input and output.
b. The ability to determine if tables, graphs, and equations represent functions.
c. The ability to determine the domain and range of functions as mathematical abstractions or in a physical context.
d. The ability to compose functions.
e. The ability to determine from the graph of a function the values of the independent variable for which the function increases, decreases, or remains constant.
3. Linear Functions. Students will demonstrate the ability to:
a. Determine when two real-world variables are related by a linear or piecewise linear function.
b. Model the behavior of two real-world variables that are directly proportional or are related by a linear or piecewise linear function using tables, graphs, equations, or combinations thereof.
c. Use a linear function to approximate the value of a non-linear function.
d. Interpret the intersection of the graphs of linear functions as equilibrium points.
e. Evaluate linear and piecewise linear functions.
f. Define, calculate, and interpret average rate of change as slope.
g. Define the linear function and the general equation of the linear function.
4. Exponential Functions. Students will demonstrate the ability to:
a. Determine when two real-world variables are related by an exponential function.
b. Model the behavior of two real-world variables that are related by an exponential function using tables, graphs, equations, or combinations thereof including such applications as population growth and decay, radioactive decay, simple and compound interest, inflation, the Malthusian dilemma, musical pitch, and the Rule of 70.
c. Change the base of an exponential function to determine rate of growth/decay, growth/decay factor, and effective and nominal interest rate.
d. Express continuous growth/decay in terms of the number e.
e. Evaluate exponential functions.
f. Determine the exponential equation model from the table or graphical model.
g. Compare linear to exponential growth.
5. Logarithmic Functions. Students will demonstrate:
a. The ability to determine when two real-world variables are related by a logarithmic function.
b. The ability to model the behavior of two real-world variables that are related by a logarithmic function using tables, graphs, equations, or combinations thereof including such applications as pH and the decibel system.
c. Their understanding of the natural logarithm.
d. The ability to graph logarithmic functions.
6. Polynomial and Quadratic Functions. Students will demonstrate the ability to:
a. Predict the shape of graphs of polynomial functions degree n.
b. Estimate horizontal intercepts of polynomial functions from their graphs.
c. Determine the horizontal intercepts of polynomial functions in factored form.
d. Determine when two real-world variables are related by a quadratic function by calculating the average rate of change of the average rates of change.
e. Model the behavior of two real-world variables that are related by a quadratic function using tables, graphs, equations, or combinations thereof including such applications as maximum area for fixed perimeter, minimum perimeter for fixed area, free fall, maximum profit, and break-even analysis.
f. Determine the vertex, axis of symmetry, and horizontal and vertical intercepts of quadratic functions in either the a-b-c or a-h-k forms.
g. Convert quadratic functions from the a-b-c form to the a-h-k form and vice versa.