Pre Calculus
Pre-Calculus bridges the gap between Algebra and Calculus by deepening students' understanding of function behavior across polynomial, rational, exponential, logarithmic, and trigonometric families. The course also covers conic sections, sequences and series, vectors, polar coordinates, and an informal introduction to limits that prepares students for the rigors of Calculus.
Who Should Take This
This course is ideal for students who have completed Algebra II or Trigonometry and are preparing to enroll in AP Calculus, college Calculus, or any STEM program requiring mathematical fluency beyond algebraic manipulation. Learners should be comfortable with function notation, equation solving, and basic graphing and should aim to master a broad toolkit of analytic techniques.
What's Included in AccelaStudy® AI
Adaptive Knowledge Graph
Practice Questions
Lesson Modules
Console Simulator Labs
Exam Tips & Strategy
13 Activity Formats
Course Outline
1Function Fundamentals and Transformations 8 topics
Describe the definition, notation, domain, range, and graph characteristics of function families: linear, quadratic, square root, cube root, absolute value, and piecewise functions
Apply vertical and horizontal shifts, reflections across axes, and vertical and horizontal stretches or compressions to transform the graph and equation of a parent function
Determine whether a function is even, odd, or neither by testing the symmetry condition f(negative x) and explain the graphical symmetry (y-axis versus origin) implied by each classification
Perform arithmetic combinations (sum, difference, product, quotient) and compositions of functions, state the domain of each combined function accounting for any new restrictions
Determine the inverse of a function algebraically by swapping x and y and solving, verify the result using composition, and state the domain and range restrictions that ensure the inverse is a function
Analyze the relationship between a function and its inverse graphically as reflections over y equals x and algebraically via the compositions f(f-inverse(x)) equals x and f-inverse(f(x)) equals x
Evaluate piecewise-defined and absolute value functions for specific input values, graph them accurately, and identify continuity and any corner points or discontinuities
Compare the rates of growth of polynomial, exponential, and logarithmic functions as x grows without bound and analyze which function family dominates in long-run behavior
2Polynomial Functions 8 topics
Describe the end behavior of a polynomial function based on its degree and leading coefficient using the notation as x approaches positive or negative infinity, f(x) approaches positive or negative infinity
Find all real zeros of a polynomial function using the Rational Zero Theorem, synthetic division, and factoring, and use the Intermediate Value Theorem to confirm roots between sign changes
Apply the Remainder Theorem and Factor Theorem to evaluate polynomials at a value via synthetic division and determine whether a given linear expression is a factor of the polynomial
Graph polynomial functions by analyzing degree, leading coefficient, zeros with their multiplicities (touch vs. cross behavior), and y-intercept, sketching the curve without a calculator
Analyze complex zeros of polynomial functions using the Fundamental Theorem of Algebra and the Complex Conjugate Pairs Theorem, determining the complete factored form over complex numbers
Evaluate rational functions by identifying vertical asymptotes (denominator zeros), horizontal or oblique asymptotes (degree comparison), holes (common factors), and x- and y-intercepts
Graph rational functions by combining asymptote analysis, intercepts, and sign charts to determine the behavior in each interval between vertical asymptotes
Analyze the graph of a polynomial to determine its minimum degree, sign of its leading coefficient, the multiplicity of each visible zero, and write a possible equation consistent with the graph
3Exponential and Logarithmic Functions 8 topics
Identify properties of exponential functions including base constraints, horizontal asymptote, domain, range, and whether the function models growth (base greater than one) or decay (base between zero and one)
Apply transformations (shifts, reflections, stretches) to exponential functions, graph the result, and write the transformed equation from a described or graphed transformation
State the properties of logarithms (product, quotient, power, change of base) and apply them to expand and condense logarithmic expressions involving multiple terms
Solve exponential equations by taking logarithms of both sides or by equating exponents when bases can be made equal, expressing solutions in exact and decimal form
Solve logarithmic equations by applying properties of logarithms to combine terms into a single logarithm, converting to exponential form, and checking solutions for extraneous roots
Apply exponential models including compound interest, continuous growth A equals P e to the rt, half-life decay, and population models to solve real-world problems and interpret parameters
Analyze the inverse relationship between exponential and logarithmic functions including domain/range swaps, graph reflections over y equals x, and cancellation identities
Solve exponential growth and decay problems requiring determination of the growth rate or initial value from two data points by setting up a system and solving for the parameters of the model
4Trigonometric Functions 8 topics
Describe the unit circle including radian and degree measures of standard angles, coordinates of key points, and the definitions of sine, cosine, and tangent as coordinates and ratios
Apply periodic, amplitude, phase shift, and vertical shift parameters to graph transformations of sine, cosine, and tangent functions from an equation of the form y equals A times f(Bx minus C) plus D
Graph the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) showing correct period, asymptotes, and key values, and identify each from its graph
Apply Pythagorean, reciprocal, and quotient trigonometric identities to simplify expressions and verify equalities by transforming one side into the other
Solve trigonometric equations over a specified interval and over the general solution by finding reference angles, applying inverse trig functions, and accounting for periodicity
Apply the inverse trigonometric functions arcsine, arccosine, and arctangent including their restricted domains, ranges, and composition rules such as sin(arcsin(x)) and arcsin(sin(x))
Analyze real-world harmonic motion and wave problems by identifying amplitude, period, phase shift, and vertical shift from physical context and writing the corresponding sinusoidal function
Apply sum-to-product and product-to-sum trigonometric identities to simplify expressions involving sums or products of sinusoids and verify these identities algebraically
5Conic Sections 7 topics
Identify the four conic sections (circle, parabola, ellipse, hyperbola) by their standard-form equations and describe the geometric definition of each as a locus of points
Write and graph parabolas in vertex form and standard form, identify the vertex, focus, directrix, and axis of symmetry, and convert between forms by completing the square
Write and graph ellipses in standard form, identify center, vertices, co-vertices, and foci using the relationship c squared equals a squared minus b squared, and sketch the ellipse
Write and graph hyperbolas in standard form, identify center, vertices, foci, and asymptotes using the relationship c squared equals a squared plus b squared, and sketch the branches
Convert a general second-degree equation Ax squared plus Bxy plus Cy squared plus Dx plus Ey plus F equals zero to standard conic form by completing the square and identify the conic type
Analyze real-world applications of conic sections including parabolic reflectors, elliptical orbits, and hyperbolic navigation by connecting the geometric properties to the physical phenomena
Solve systems of equations involving conic sections by substituting one equation into the other to find the coordinates of intersection points and interpret these geometrically
6Sequences and Series 7 topics
Identify arithmetic and geometric sequences by finding the common difference or common ratio, and state the explicit and recursive formulas for the nth term of each type
Calculate the sum of a finite arithmetic series using the formula S sub n equals n over 2 times (a sub 1 plus a sub n) and the finite geometric series formula S sub n equals a sub 1 times (1 minus r to the n) over (1 minus r)
Apply sigma notation to write series compactly, evaluate finite sums using summation formulas, and convert between sigma notation and expanded sum form
Determine whether an infinite geometric series converges or diverges based on the absolute value of the common ratio and calculate the sum of a convergent infinite geometric series
Apply the Binomial Theorem and Pascal's Triangle to expand (a plus b) to the power n, identifying binomial coefficients and the general term using combination notation n choose r
Analyze annuity and loan repayment scenarios by modeling them as finite geometric series and computing future value, present value, or payment amounts
Apply the Principle of Mathematical Induction to verify formulas for integer sums and sequences by completing the base case and inductive step in a formal proof
7Vectors and Polar Coordinates 7 topics
Describe vectors in component form and as directed line segments, state properties including magnitude, direction angle, and unit vector, and distinguish vectors from scalars
Perform vector addition, scalar multiplication, and subtraction both graphically (tip-to-tail, parallelogram) and algebraically using component-wise operations
Calculate the dot product of two vectors and use it to find the angle between them, determine whether they are orthogonal, and compute the projection of one vector onto another
Convert points and equations between rectangular (Cartesian) and polar coordinate systems using the relationships x equals r cosine theta, y equals r sine theta, and r squared equals x squared plus y squared
Graph polar equations including circles, rose curves, limacons, and lemniscates by plotting points or using symmetry tests and technology, identifying key features of each curve
Analyze force, velocity, or navigation problems involving vector addition by resolving vectors into components, computing the resultant, and interpreting magnitude and direction in context
Graph parametric equations by creating a table of (x(t), y(t)) values, plotting the curve with direction of motion indicated, and eliminating the parameter to write an equivalent rectangular equation
8Introduction to Limits 7 topics
Describe the informal concept of a limit as the value a function approaches as the input approaches a given value, distinguishing the limit from the function value at that point
Estimate limits numerically using tables of values and graphically by tracing the function, identifying cases where left-hand and right-hand limits agree, differ, or are infinite
Calculate limits algebraically using direct substitution, factoring and cancellation, rationalization, or recognizing indeterminate forms and applying limit laws
Apply limit concepts to identify removable discontinuities (holes), jump discontinuities, and infinite discontinuities in rational and piecewise functions
Calculate limits at infinity for rational functions by comparing degrees of numerator and denominator to determine horizontal asymptotes and end behavior
Analyze how the limit concept connects to continuity, the Intermediate Value Theorem, and the informal definition of the derivative as a difference quotient, bridging pre-calculus to Calculus
Evaluate the limit of the difference quotient (f(x plus h) minus f(x)) over h as h approaches zero for polynomial and simple rational functions as an introduction to instantaneous rate of change
Scope
Included Topics
- Function families (polynomial, rational, exponential, logarithmic, trigonometric) and their graphs, transformations of functions (shifts, stretches, reflections), inverse functions (existence, finding, graphing), polynomial functions (end behavior, zeros, multiplicity, the Remainder and Factor theorems), rational functions (asymptotes, holes, end behavior), exponential and logarithmic functions (properties, equations, applications to growth/decay), trigonometric functions (unit circle, graphs, equations, identities), conic sections (parabola, ellipse, hyperbola — equations and graphs), sequences and series (arithmetic, geometric, sigma notation, finite sums), vectors in two dimensions (operations, magnitude, direction, dot product), polar coordinates and polar graphs, introduction to limits
Not Covered
- Formal epsilon-delta definition of limits
- Differentiation and integration (covered in Calculus)
- Linear algebra beyond two-dimensional vectors
- Probability and statistics
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