A2: Algebra II with Statistics

Algebra II with Statistics

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Complex numbers, polynomial identities and operations, and rational expressions form the algebra core. Trigonometric functions, identities, and the law of sines and cosines connect to periodic modeling. Matrices, statistical inference, and data analysis round out the course.

A2.1Identify numbers written in the form a + bi, where a and b are real numbers and i^2 = –1, as complex numbers.A2.10Create equations and inequalities in one variable and use them to solve problems. Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.A2.11Solve quadratic equations with real coefficients that have complex solutions.A2.12Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.A2.13Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.14Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).A2.15Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend to polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.A2.16Identify the effect on the graph of replacing f(x) by f(x) + k, k ∙ f(x), f(k ∙ x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.17For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; and periodicity. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.18Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.19Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.2Use matrices to represent and manipulate data.A2.20Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.A2.21Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle, building on work with non-right triangle trigonometry.A2.22Use the mathematical modeling cycle to solve real-world problems involving polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions, from the simplification of the problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution’s feasibility.A2.23Use mathematical and statistical reasoning about normal distributions to draw conclusions and assess risk; limit to informal arguments.A2.24Design and carry out an experiment or survey to answer a question of interest, and write an informal persuasive argument based on the results.A2.25From a normal distribution, use technology to find the mean and standard deviation and estimate population percentages by applying the empirical rule.A2.26Describe the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.A2.27Distinguish between a statistic and a parameter and use statistical processes to make inferences about population parameters based on statistics from random samples from that population.A2.28Describe differences between randomly selecting samples and randomly assigning subjects to experimental treatment groups in terms of inferences drawn regarding a population versus regarding cause and effect.A2.29Explain the consequences, due to uncontrolled variables, of non-randomized assignment of subjects to groups in experiments.A2.3Multiply matrices by scalars to produce new matrices.A2.30Evaluate where bias, including sampling, response, or nonresponse bias, may occur in surveys, and whether results are representative of the population of interest.A2.31Evaluate the effect of sample size on the expected variability in the sampling distribution of a sample statistic.A2.32Produce a sampling distribution by repeatedly selecting samples of the same size from a given population or from a population simulated by bootstrapping (resampling with replacement from an observed sample). Do initial examples by hand, then use technology to generate a large number of samples.A2.33Use data from a randomized experiment to compare two treatments; limit to informal use of simulations to decide if an observed difference in the responses of the two treatment groups is unlikely to have occurred due to randomization alone, thus implying that the difference between the treatment groups is meaningful.A2.34Define the radian measure of an angle as the constant of proportionality of the length of an arc it intercepts to the radius of the circle; in particular, it is the length of the arc intercepted on the unit circle.A2.35Choose trigonometric functions (sine and cosine) to model periodic phenomena with specified amplitude, frequency, and midline.A2.36Prove the Pythagorean identity sin^2 (θ) + cos^2 (θ) = 1 and use it to calculate trigonometric ratios.A2.37Derive and apply the formula A = ½·ab·sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side, extending the domain of sine to include right and obtuse angles.A2.38Derive and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Extend the domain of sine and cosine to include right and obtuse angles.A2.4Add, subtract, and multiply matrices of appropriate dimensions.A2.5Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.A2.6Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.A2.7Prove polynomial identities and use them to describe numerical relationships.A2.8Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. Extend to radical equations.A2.9For exponential models, express as a logarithm the solution to ab^ct = d, where a, c, and d are real numbers and the base b is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.
Example Problems


What is the real part of x?
Solve the equation:
Solve:
Evaluate:
The parabola is reflected across the -axis.

What is the equation of the new parabola?
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