Algebra and Trigonometry Study Guide
7 units, each with what to know, how to apply it, the mistakes that cost marks, and a linked practice set.
Seven units
Four function-family units followed by three trigonometry units.
Combined course
Covers Algebra 2 function work and a full introduction to trigonometry in one year.
Prerequisite
Algebra 1 and Geometry.
Where it leads
Precalculus or AP Precalculus, with the trigonometry already established.
Units
The 7 units
The first half establishes function families and the second applies the same analytical habits to periodic functions. Students who treat trigonometry as a new subject rather than another family find the second half harder than it needs to be.
Functions, Transformations, and Inverses
Function notation, domain and range, and the transformations that move any graph. Learned once, this unit applies to every family that follows.
Quadratic and Polynomial Functions
Zeros, multiplicity, and end behavior. Together these three facts determine the shape of any polynomial graph before a single point is plotted.
Radical and Rational Functions
Two families defined by what they exclude. Radicals restrict the domain; rationals introduce asymptotes and holes, and both create extraneous solutions.
Exponential and Logarithmic Functions
Growth by constant ratio, and the inverse that recovers the exponent. Most of the unit follows from those two sentences.
Unit Circle and Trigonometric Functions
Extending trigonometry beyond right triangles. Structure and symmetry replace memorisation, which is what makes unfamiliar angles tractable.
Trigonometric Graphs and Models
Four parameters that describe any sinusoid, and using them to model anything that repeats.
Identities and Trigonometric Equations
Rewriting trigonometric expressions into a form that can be evaluated or solved, and reporting every solution the interval demands.
Habits
The habits that run through every unit
Four habits this sequence assesses directly, whichever unit the question comes from.
1. Write what you were given
List the known quantities, the quantity asked for, and any restriction on the domain before starting. Most lost marks come from answering a slightly different question.
2. Choose a representation
Equation, graph, table, or diagram are four views of one relationship. When one stalls, switch; the assessment usually rewards moving between them.
3. Justify the step, not the answer
Name the property or theorem that licenses each move. On written work the justification is frequently worth more than the arithmetic.
4. Test the boundary
Zero, a negative value, an excluded value, an endpoint. A method that works in the middle of a range fails at its edges, and that is where questions are set.
Plan
How to use this guide
Carry function habits into trigonometry
Domain, transformation, and inverse questions work the same way for sine as for a polynomial. Treating trigonometry as separate makes it harder.
Build the unit circle from structure
Special triangles and symmetry survive into unfamiliar angles; a memorised table does not.
Decide the quadrant before the value
Nearly every trigonometry error at this level is a sign error, and this habit removes most of them.
Factor the argument before reading a shift
Phase shift is the single most misread parameter in the graphing unit.
Multiple choice
- • Identify what kind of object the question is about before computing; the structure usually names the method.
- • Check for excluded values whenever a denominator, an even root, or a logarithm appears.
- • When two options differ only at an endpoint, test that endpoint rather than re-reading the question.
- • Estimate the size of the answer first so an implausible result is caught immediately.
- • Watch for questions asking which statement is true rather than for a value; those are read carelessly most often.
Written work
- • Show the steps that carry the reasoning, not every line of arithmetic.
- • Name the property, theorem, or identity that justifies the key move.
- • Give units and context in the final sentence when the problem had them.
- • State any restriction you imposed and why.
- • Check the answer back in the original statement before moving on.
Scope
What this course covers
Core
- • Functions, transformations, composition, and inverses
- • Polynomial and rational function behavior
- • Exponential and logarithmic functions
- • The unit circle and exact values
- • Sinusoidal graphs and periodic modeling
- • Identities and trigonometric equations
Breadth
- • Triangle applications of trigonometry
- • Modelling repeating phenomena
- • Solving equations that have no exact algebraic route
Varies by course
- Polar coordinates. Included in some versions of this course.
- Vectors. Occasionally attached to the trigonometry half.
- Sequence order. Some sections teach trigonometry first.