# OUTCOMES MATHEMATICAL ANALYSIS/TRIGONOMETRY 1, 2

Each outcome is preceded by the phrase : The intent of
instruction is that

most students will:

1. Review of Real Numbers , The Plane, and Algebra.

1.1 The Real Number System.

a. identify subsets of real numbers.

b. know the properties of real numbers.

c. demonstrate understanding of the order properties.

d. know and use the notation for naming intervals on a number line.

e. know and use the definition of absolute value.

1.2 Exponents and Radicals.

a. know and use properties of exponents.

b. use calculators to evaluate exponential expressions.

c. know and use the properties of radicals.

d. evaluate and simplify radicals.

1.3 Polynomials and Factoring.

a. find the sums, differences, and products of polynomials.

b. factor polynomials .

1.4 Fractional Expressions

a. state the domain of an algebraic expression.

b. factor polynomials.

1.5 Solving Equations

a. solve an equation with fractional parts.

b. solve quadratic equations .

c. solve equations involving radical expressions.

d. solve equations with absolute values.

1.6 Solving Inequalities.

a. solve linear and absolute value inequalities.

b. solve quotient and high order inequalities.

2. Functions and Their Graphs.

2.1 The Cartesian Plane

a. review the terminology of the Cartesian plane.

b. know and use the distance and midpoint formulas.

2.2 Graphs of Equations.

a. use the point plotting method to graph equations.

b. identify x and y intercepts and use them in graphing.

c. identify and use symmetry to graph equations.

d. apply the distance formula to develop the circle equation.

2.3 Lines in a Plane.

a. find the slope of a line.

b. write and graph linear equations in several forms.

c. use the relationship between parallel and perpendicular lines.

2.4 Functions.

a. define function and find domain and range.

b. given f(x) and a, find f(a).

2.5 Graphs of Functions.

a. find the domain and range of a function from its graph.

b. determine when a function is increasing, decreasing, or a constant.

c. use the test for odd and even functions.

d. identify and use the basic types of transformations of a graph.

2.6 Combinations of Functions.

a. find the sum, difference, and quotient of functions.

b. find the composition of two functions.

2.7 Inverse Functions.

a. apply the test to determine if a function is one-to-one.

b. find the inverse of a function if it exists.

2.8 Variation.

solve problems using direct and inverse variation.

3. Polynomials and Rational Functions

3.1 Quadratic Functions.

a. know the definition of a polynomial function and of a quadratic

function.

b. know the standard form of a quadratic function.

c. find the maximum or minimum value of a quadratic function.

3.2 Polynomial Functions.

a. Use the leading coefficient to determine right and left behavior of a

graph.

b. find the real zeroes of a function.

c. know and use the intermediate value theorem.

d. write a polynomial function given its zeroes.

e. graph a polynomial function.

f. approximate zeroes of a function.

3.3 Polynomial Division.

a. use synthetic division to evaluate polynomials at a given value.

b. use synthetic division to apply the remainder and factor theorems.

3.4 Real Zeros of Polynomial Functions.

a. use Descarte’s Rule of signs to analyze polynomial functions.

b. apply the rational zero test to find zeroes of higher degree

polynomials.

c. find the upper and lower bound for the zeroes of a polynomial.

3.5 Complex Numbers .

a. know the definition of a complex number.

b. add, subtract, and multiply complex numbers.

c. use complex conjugates to divide complex numbers.

d. solve quadratic equations yielding complex solutions.

3.6 The Fundamental Theorem of Algebra.

a. find all zeroes of a function and write the polynomial as a product of

linear factors.

b. write higher order equations given their roots.

3.7 Rational Functions.

a. find intercepts, domain and asymptotes of rational functions.

b. graph rational functions.

4. Exponential and Logarithmic Functions .

4.1 Exponential Functions.

a. recognize an exponential function and know the characteristics of its

graph.

b. solve exponential equations.

c. solve problems which use properties of exponential functions

including the number e.

4.2 Logarithmic Functions.

a. define and evaluate logarithms.

b. convert between logarithmic and exponential forms.

c. know the definition for common and natural logarithms.

d. find the domain and sketch the graph of logarithmic functions.

4.3 Properties of Logarithms.

a. know and use the change of base formula for logarithms.

b. know and use the properties of logarithms.

4.4 Exponential and Logarithmic Applications.

apply exponential and logarithmic models to solve many types of

stated problems.

5. Trigonometry

5.1 Radian and Degree Measure.

a. understand the terminology of angles.

b. convert between degree and radian measurement.

c. find arc length.

5.2 Trigonometric Functions and the Unit Circle.

a. define the six trig functions in terms of the unit circle.

b. evaluate trig functions of special numbers.

5.3 Trigonometric Functions and Right Triangles.

a. define the six trig functions in terms of a right angle.

b. evaluate trig functions of special numbers.

c. know the basic 8 fundamental identities.

d. use the right triangle to evaluate trigonometric functions.

e. know the sine, cosine and tangent of special angles.

5.4 Trigonometric Functions of Any Angle.

a. know the signs of the trigonometric functions in each quadrant.

b. use reference angles and the period of a function to find values of

trigonometric functions of angles.

5.5 Graphs of Sine and Cosine Functions.

a. graph the basic sine and cosine functions.

b. identify amplitude, period, and phase shift of sine and cosine.

c. graph translations of sine and cosine functions.

d. write the equation for the sine or cosine function given a graph or

information about the graph.

5.6 Graphs of Other Trigonometric Functions.

graph the tangent, cotangent, secant, and cosecant.

5.7 Graphs of Combination Functions.

graph by addition or ordinates .

5.8 Polar coordinates(11.6)

a. plot points in a polar coordinate system.

b. convert points and equations from rectangular system to polar

system and from polar to rectangular.

5.9 Graphs of Polar Equations (11.7)

graph polar equations.

5.10 Inverse Trigonometric Functions (5.8)

a. know the definition of inverse sine, cosine, and tangent functions.

b. know the domain and range of the three inverse functions.

c. evaluate the inverse functions with and without a calculator.

5.11 Applications of Trigonometry(5.9)

a. solve for missing parts of a triangle.

b. draw illustrations and solve right triangle application problems.

6. Analytic Trigonometry.

6.1 Fundamental Identities.

know and use the fundamental identities.

6.2 Verifying Trigonometric Identities.

use the fundamental identities to verify equalities.

6.3 Trigonometric Equations.

apply identities and algebraic methods to solve trigonometric

equations.

6.4 Sum and Difference Formulas.

know and use the sum and difference formulas for sine, cosine, and

tangent functions.

6.5 Multiple Angle Formulas

know and use the double angle and half-angle formulas.

7. Additional Topics in Trigonometry

7.1 Law of Sines

a. solve oblique triangles when the law of sines applies.

b. determine when data given describes none, one or two triangles.

c. find the area of oblique triangles.

7.2 Law of Cosines

a. solve oblique triangles when the law of cosines applies.

b. use Heron’s formula to find the area of oblique triangles.

7.3 Trigonometric Form of a Complex Number (7.4)

a. graph complex numbers in a complex plane.

b. find the absolute value of a complex number.

c. write complex numbers in standard and trigonometric form.

d. find the product and quotient of complex numbers in trigonometric

form.

7.4 DeMoivre’s Theorem (7.5)

a. use DeMoivre’s Theorem to find powers of complex numbers.

b. find the nth roots of a complex number.

8. Topics in Analytic Geometry.

8.1 Lines (11.1).

a. find the inclination of a line.

b. find the angle between two lines.

c. find the distance between a point and a line.

8.2 Parabolas (11.2)

a. know and use the definition and standard equation of a parabola.

b. given the equation of a parabola, find the directrix, focus and axis.

c. write the equation of a parabola given specific properties.

d. graph an ellipse given an equation.

8.3 Ellipses (11.3)

a. know and use the definition and standard form of an ellipse.

b. use the equation of an ellipse to find its vertices, foci, eccentricity,

and axes.

c. write the equation of an ellipse given specific properties.

d. graph an ellipse given an equation.

8.4 Hyperbolas (11.4)

a. know and use the definition and standard form of a hyperbola.

b. use the equation of the hyperbola to find its vertices, foci,

eccentricity, asymptotes, and axes.

c. write the equation of a hyperbola given specific properties.

d. graph a hyperbola given an equation.

Systems of Quadratics (Optional)

Sequences and Series

9.1 Sequences and Summation Notation (10.1)

a. know the definition of sequence and series.

b. understand and use sigma notation.

c. write an expression for the nth term of a sequence.

9.2 Arithmetic Sequences (10.2)

a. know and use the definition and formulas for arithmetic sequences

and series.

b. find arithmetic means.

c. solve application problems using sequences and series.

9.3 Geometric Sequences.

a. know and use the definition and formula for geometric sequences

and series.

b. solve application problems using sequences and series.

9.4 Mathematical Induction.

know and use the principle of math induction.

9.5 Binomial Theorem

a. expand a binomial using the theorem.

b. find a specific term of a binomial expansion using the theorem.

Optional Topics (Time Permitting)

Limit

Matrices

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