Algebra 1 begins with learning to translate verbal phrases into symbols. This leads to the topic of formulas and equations. In particular, proportions are solved and linear and quadratic equations are solved and graphed. Along the way, factoring polynomials and properties of square roots are introduced.
It is customary to include some introductory Geometry topics, such as the Pythagorean theorem. Additionally, Probability is introduced, including Permutations and Combinations, and an introduction to Statistics.
The quadratic formula is presented, along with an introduction to complex numbers.
The laws of exponents are extended to the cases of zero, negative and fractional exponents.
The idea of a function and its inverse is introduced.
Extensive use is made of exponential and logarithmic functions, including graphing and solving equations. Applications include compound interest problems and radioactive decay.
Additional topics include the geometric series and the binomial theorem.
The derivative is presented in terms of the instantaneous rate of change and as the slope of the tangent to a curve. The integral is presented as an area function. The fundamental theorem of Calculus, relating derivatives and integrals, is demonstrated by analyzing the rate of change (derivative) of the area function (integral).
Techniques of differentiation are introduced, with applications to maximum/minimum problems and curve sketching. Integrals are applied to calculating volume as well as area. Derivatives and integrals of trigonometric, exponential, and logarithmic functions are studied, with an introduction to differential equations.
Special integration techniques, including substitution, integration by parts and partial fractions are studied.
After reviewing the basic concepts of Geometry, including lines, angles, and triangles, the focus of the course is proof. An essential concept is that of congruence, and methods of proving that triangles are congruent are introduced.
Applications are made to quadrilaterals, including parallelograms, rectangles, squares and trapezoids.
The concept of similar triangles is introduced. leading to a proof of the Pythagorean theorem.
The geometry of the circle and an introduction to three dimensional geometry is included.
Additionally, coordinate geometry is reviewed, and then applied as a proof method, with frequent use of transformation geometry.
Precalculus prepares students for a first course in Calculus, as well as introducing topics that will be needed in other Mathematics courses.
In preparation for a first course in Calculus, polynomial and rational functions are graphed, conic sections are analyzed, and limits are illustrated with the tangent line and area problem.
In preparation for Linear Algebra, systems of linear equations are analyzed using matrices and determinants, and three dimensional coordinate geometry is introduced, including vectors.
In preparation for the further study of Algebra, the representation of complex numbers using polar coordinate is presented, and the fundamental theorem of Algebra is introduced, illustrated using DeMoivre's formula for the nth roots of unity.
The trigonometry of the right triangle is introduced, including the basic functions sine, cosine and tangent and their cofunctions. Radian measure of angles is presented, and the trigonometric functions are defined for arbitrary angles. The trigonometric functions are graphed, with a discussion of their amplitude, period and phase shift. Inverse trigonometric functions are introduced.
The laws of cosine and sine are proved and applied to solving triangles.
Basic trigonometric identities are proved, including half angle and double angle identities.