Pre-Calculus

Scope & Sequence

Semester 1 

Semester 2 

 
Chapter 1: Functions 

  • 1.1 Subsets of real numbers, finding domain algebraically 

  • 1.2 Functions, their parts, domain/range, symmetry 

  • 1.3 Continuity and end behavior via limits 

  • 1.4 Increase/decrease/constant intervals, extrema, average rate of change 

  • 1.5/3.1/3.2 Identify, graph, describe parent functions and transformations 

 

 
Chapter 2: Power, Polynomial, and Rational Functions 

  • 2.1 Solve radical equations, extraneous solutions 

  • 2.2 Polynomial graphs, end behavior, multiplicity of zeros 

  • 2.3 Divide polynomials (long, synthetic), Remainder and Factor Theorems 

  • 2.4 Find zeros of polynomials (real/complex) 

  • 2.5 Features of rational functions (asymptotes, holes), solve rational equations 

 

Chapter 4: Trigonometric Functions 

  • 4.1 The 6 trig ratios, right triangle trig 

  • 4.2 Radians and degrees 

  • 4.3 Trig values in radians/degrees, unit circle reference and coterminal angles 

  • 4.4 Graph transformations of sine and cosine, sinusoidal modeling 

  • 4.5 Parent functions of tangent and reciprocal trig functions (omit transformations) 

  • 4.6 Evaluate inverse trig functions (show but do not assess graphs of these) 

  • 4.7 Law of Sines, Law of Cosines, areas of oblique triangles 

 
Chapter 5: Trigonometric Identities and Equations 

  • 5.1 Identity/use basic trig identities to simplify, rewrite, and evaluate (quotient, reciprocal, Pythagorean, cofunction, even/odd) 

  • 5.2 Verify trig identities 

  • 5.3 Solve trig equations using algebra and basic identities 

  • 5.4 Use sum/difference identities to evaluate and solve 

  • 5.5 Use double-angle identities to evaluate and solve 

 

Chapter 7: Conic Sections and Parametric Equations 

  • 7.1 Graph and write equations of parabolas, identify the vertex, focus, axis of symmetry, and directrix 

  • 7.2 Graph and write equations of ellipses (including by completing the square), identify center, vertices, covertices, foci, eccentricity 

  • 7.3 Graph and write equations of hyperbolas, identify characteristics as in 7.2 

 

 
Chapter 12: Limits (optional/as time allows) 

  • 12.1 Estimate limits at a point and limits at infinity 

  • 12.2 Compute limits at a point and limits at infinity (includes direct substitution and dividing out) 

 

Essential Learnings

Semester 1 

Ch.1 Functions from a Calculus Perspective 

1.1  

Subsets of real numbers.  

Identity and evaluate functions (including piecewise-defined functions). 

Find domains of function algebraically (including rational and radical functions). 

Domains of exponential and logarithmic functions. 

Evaluating exponential and logarithmic expressions. 

1.2  

Use graphs of functions to estimate function values and find domains, ranges, y-intercepts, and zeros of functions.   

Explore symmetries of graphs, and identify even and odd functions. 

1.3 

Use limits to determine the continuity of a function.  

Classify discontinuities from graphs and tables. 

Use limits to describe end behavior of functions. 

1.4 

Determine intervals of which functions are increasing, constant, or decreasing, and determine maxima and minima of functions.  

Determine the average rate of change of a function. 

1.5, 3.1, 3.2 

Identify, graph, and describe parent functions (constant, identity, quadratic, cubic, square root, reciprocal, absolute value, and greatest integer function).  

Insert: 3.1 and 3.2 Parent functions of exponential and logarithmic functions.  

Identify and graph transformations of functions.  

Optional: Add in parent functions for sine and cosine as well. 

1.7 

Use graphs of functions to determine if they have inverse functions.  

Find inverse functions graphically and algebraically.   

Verify algebraically two functions are inverse functions. 

Insert: Exponentials and Logarithms as inverses. 

2.1 

Solve radical equations. 

Includes sum of radicals equal to a value or linear function. 

Includes extraneous solutions. 

Optional: Graph and analyze power functions. 

Optional: Graph and analyze radical functions. 

Optional: Power Regression Ex. #4 

3.3/3.4  

Optional: Properties of Logarithms. (3.3 Ex #1-4 only) 

Optional: Solving exponential equations. (3.4 Ex #1-4 only) 

2.2 

Graph polynomial functions. 

Leading term test for end behavior. 

Repeated zeros of polynomial functions. 

Optional: Polynomial Regression Ex. #7 

2.3 

Divide polynomials using long division and synthetic division. 

Use the Remainder and Factor Theorems. 

Optional: Descartes Rules of Signs. 

2.4 

Find real zeros of polynomial functions. 

Solving using u-substitution (higher degree polynomials in quadratic forms. 

Find complex zeros of polynomial functions – maybe add in some review 

2.5 

Analyze and graph rational functions. Features include vertical and horizontal asymptotes, oblique asymptotes, and holes. 

Solve rational equations. 

3.1  

Optional: Calculate future values of annuities and monthly payments pg. 170 

4.1 

Ratios of 6 trigonometric functions. 

Find values of trigonometric functions for acute angles of right triangles. 

Trigonometric values of special right triangles. 

Solve right triangles using a calculator. 

4.2 

Convert degree measures of angles to radian measures, and vice versa. 

Arc length, and area of a sector. 

Angular and linear speeds. 

Convert degrees to degrees-minutes-seconds 

4.3 

Find values of trigonometric functions for any angle in radians and degrees (quadrantal angles included). 

Reference and coterminal angles. 

Find values of trigonometric functions using the unit circle. 

4.4 

Graph transformations of the sine and cosine functions. 

Model real world behavior with sinusoidal functions. 

Sinusoidal Regression Ex. #7 

4.5 

Parent functions of tangent and reciprocal trigonometric functions. 

Graph transformations tangent and reciprocal trigonometric functions. 

4.6 

Evaluate inverse trigonometric functions. 

Find compositions of trigonometric functions. 

Graphs of inverse trig functions. – show but don’t make them do the graphing  

4.7 

Solve oblique triangles by using the Law of Sines or the Law of Cosines. 

Find areas of oblique triangles. 

 

Semester 2 

Ch. 5 Trigonometric Identities and Equations 

5.1 

Identify and use basic trigonometric identities to find trigonometric values (quotient, reciprocal, Pythagorean, cofunction, even and odd). 

Use basic trigonometric identities to simplify and rewrite trigonometric expressions. 

5.2 

Verify trigonometric identities. 

5.3 

Solve trigonometric equations using algebraic techniques. 

Solve trigonometric equations using basic identities. 

Optional: Real World Ex #4 

5.4 

Use sum and difference identities to evaluate trigonometric functions. 

Use sum and difference identities to solve trigonometric equations. 

Optional: Trigonometric Functions of Multiple Angles Ex #4 

5.5 

Use double-angle identities to evaluate trigonometric expressions and solve trigonometric equations. 

Use half-angle identities to evaluate trigonometric expressions and solve trigonometric equations. 

Using power reducing properties. 

7.1 

Analyze and graph parabolas. Include vertex, focus, axis of symmetry, and directrix. 

Write equations of parabolas given characteristics. 

Optional: Tangent line to a parabola Ex #5 

7.2 

Analyze and graph ellipses and circles. Include center, vertices, covertices, and foci. 

Write equations of ellipses and circles given characteristics. 

Calculate eccentricity. 

Identify conic sections by completing the square. 

7.3 

Analyze and graph hyperbolas. Include center, vertices, covertices, foci, and asymptotes. 

Write equations of hyperbolas given characteristics. 

Optional: Estimate limits at a point. 

Optional: Estimate limits at infinity. 

Optional: Compute limits at a point. Includes direct substitution and dividing out. 

Optional: Compute limits at infinity.