Precalculus

Scope and Sequence

Students will be able to:

  • Review 6 trigonometric functions of right triangle trigonometry in terms of ratios of sides.
  • Evaluate other trigonometric functions given a trigonometric ratio.
  • Basic identities – Reciprocal, Pythagorean, cofunction
  • Identify angles in standard position
  • Identify reference angles for obtuse/non-acute angles
  • Use angles as a measure of rotation
  • Identify coterminal angles
  • Find the arc length subtended by a central angle
  • Find the area of a sector subtended by a central angle
  • Demonstrate the development of radians as a measure of arc subtended by a central angle
  • Convert degrees to radians
  • Convert radians to degrees
  • Find arc length given radians
  • Find sector area given radians
  • Evaluate trigonometric functions of 30°, 45°, and 60°.
  • Evaluate the trigonometric functions of 30°, 45°, and 60° from triangles in the first quadrant of the unit circle.
  • Find the measure of reference angles.
  • Use the trigonometric values identified in Quadrant 1 to identify the trig values for 120°,135°,150°,210°,225°,240°,300°,315°, and 330°.
  • Use the unit circle to identify trigonometric values for
  • Identify Quadrantal Angles.
  • Convert all unit circle angles to radian measure.
  • Connect radian measure to the number line.
  • Find coterminal angles using radians.
  • Find reference angles using radians.
  • Graph the cosine and sine measures for unit circle angles along a number line.
  • Define a periodic function.
  • Determine the period of sine and cosine.
  • Evaluate trigonometric functions of unit circle values without a calculator.
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Suggested Activities

Students will be able to:

  • Identify the roots for sine and cosine.
  • Analyze the graphs of sine and cosine to determine if the functions are even or odd or neither.
  • Analyze the graphs of sine and cosine to determine maximums and minimums.
  • Analyze the graphs of sine and cosine to identify regions where the function is increasing or decreasing.
  • Determine if the graphs of sine and cosine are bounded.
  • Graph sine and cosine using function transformations.
  • Define amplitude in terms of function transformations.
  • Identify function transformations that modify the period of sine and cosine.
  • Identify and apply horizontal and vertical reflections of sine and cosine.
  • Apply horizontal and vertical translations of sine and cosine.
  • Generate the function given a transformed sine or cosine graph.
  • Using the unit circle, generate all the tangents for the given angles.
  • Compare and contrast the period for tangent.
  • Graph the tangent values on a number line.
  • Express the period for tangent functions.
  • Identify any discontinuities for the tangent graph and express as repeating in terms of n.
  • Identify the location of any roots for tangent.
  • Graph tangent using function transformations.
  • Graph reciprocal trigonometric functions along a number line.
  • Determine the period for the reciprocal trigonometric functions.
  • Identify any discontinuities and express as repeating in terms of n.
  • Analyze the graphs of the reciprocal trigonometric functions to identify similarities with the graphs of sin and cosine.
  • Use the definition of inverse functions to generate the graphs for arcsin and arccos on a number line.
  • Determine the restrictions required to make trigonometric functions and their inverse functions one to one.
  • Use an inverse trig function graph to determine an angle measure from a given ratio.

Suggested Activities

Students will be able to:

  • Solve simple harmonic motion problems.
  • Identify a dampened trigonometric function graph.
  • Define angular velocity as rotational speed.
  • Convert rotations to degrees and radians.
  • Compare angular and linear velocity.
  • Express linear velocity in terms of angular velocity.
  • Derive Law of Sines
  • Apply Law of Sines
  • Determine how to identify the ambiguous case.
  • Derive Law of Cosines.
  • Apply the Law of Cosines.
  • Use arc length formulas to derive conversion factor for nautical miles compared to statute miles.
  • Determine direction of a vessel using directional bearings by sea.
  • Use Heron’s formula to determine area.
  • Determine direction of a craft by air using air navigation bearings.

Suggested Activities

Students will be able to:

  • Review basic trigonometric identities, reciprocal trigonometric functions, cofunctions, even/odd functions and pythagoran functions.
  • Prove an algebraic identity by finding a common denominator.
  • Prove trigonometric identities using a graphing calculator.
  • Factor functions composed of trigonometric functions.
  • Verify trigonometric identities algebraically by manipulating a single side of the identity to transform it into the other.
  • Finding the general solution to a trigonometric equation to identify all periodic solutions.
  • Use inverse trigonometric functions to identify all possible solutions to a trigonometric equation.
  • Graphically verify the formulas for the trigonometric values of sums and differences of angles.
  • Evaluate trigonometric expressions using the sum and difference formulas.
  • Derive/prove double angle formulas using sum and difference formulas and trigonometric identities.
  • Evaluate trigonometric expressions using double angle formulas.
  • Derive/prove power reducing formulas using sum and difference formulas and trigonometric identities.
  • Evaluate trigonometric expressions using power reduction formulas.
  • Evaluate trigonometric expressions using half-angle formulas.
  • Evaluate trigonometric expressions using product to sum formulas.
  • Solve a trigonometric equation using multiple angle formulas.

Suggested Activities

Students will be able to:

  • Define vectors as directed line segments.
  • Determine when two vectors are equivalent.
  • Find the sum of vectors graphically by tail to head representation and by parallelogram method.
  • Find the scalar product of a vector.
  • Find the magnitude of a vector.
  • Find the direction angle of a vector.
  • Write vectors in component form.
  • Determine the direction of a given vector.
  • Find the sum of vectors in component form algebraically.
  • Define Unit Vectors.
  • Find Unit vectors in the direction of a given vector.
  • Express a given vector as the sum of scalar products of unit vectors.
  • Find the dot product of two vectors.
  • Use the dot product to identify the angle between two vectors.
  • Define orthogonal vectors.
  • Determine when two vectors are orthogonal.

Suggested Activities

Students will be able to:

  • Define parametric equations and identify situations where they might be encountered.
  • Graph parametric equations using technology.
  • Generate a table of values to graph parametric equations on the coordinate plane.
  • Identify the direction of movement of a parametric variable along a graph on the coordinate plane.
  • Convert parametric equations to rectangular equations by eliminating the parametric variable.
  • Convert rectangular equations to parametric by defining a parameter.
  • Describe projectile motion using parametric equations.
  • Simulate horizontal motion using parametric equations.
  • Simulate vertical motion using parametric equations.

Suggested Activities

Students will be able to:

  • Define polar coordinates as angular movement followed by displacement.
  • Plot polar coordinates on a cartesian plan.
  • Plot polar coordinates on polar graph.
  • Determine all possible polar coordinates for a given point in rectangular coordinates.
  • Convert between rectangular and polar coordinates.
  • Graph polar equations on a coordinate plane.
  • Convert rectangular equations to polar form.
  • Convert polar equations to rectangular form.
  • Identify common polar graphs including circle, rose curves, lemniscate, cardioid, and limacon.
  • Identify equations for common polar graphs.
  • Graph polar forms using transformations.

Suggested Activities

Students will be able to:

  • Name and identify circles, ellipses, parabolas and hyperbolas from the general form equation.
  • Describe the origin of each conic section as sliced from a double napped cone.
  • Define each conic section as a locus of points.
  • Identify the focus and directrix of both horizontal and vertical parabolas from a graph.
  • Identify the focus and directrix of both horizontal and vertical parabolas from an equation.
  • Write the equation for a parabola given a graph.
  • Graph a horizontal parabola given an equation.
  • Explain the definition of an ellipse in comparison to the definition of a circle.
  • Determine whether a general equation represents an ellipse.
  • Determine whether a conic section is an ellipse or a circle.
  • Find the focal axis, foci, vertices, major and minor axes, and semimajor and semiminor axes of an ellipse from a graph.
  • Find the focal axis, foci, vertices, major and minor axes, and semimajor and semiminor axes of an ellipse from an equation in standard form.
  • Determine the equation of an ellipse given its foci and major/minor axis and center.
  • Define the eccentricity of an ellipse and describe its effect on the figures shape.
  • Find the eccentricity of an ellipse given a graph.
  • Find the eccentricity of an ellipse given an equation in standard form.
  • Convert the general equation of an ellipse to standard form.
  • Express the definition of a hyperbola as locus of points.
  • Explain the definition of a hyperbola, making a connection between the locus definition and the conic equation.
  • Determine whether a function is a hyperbola given a graph.
  • Determine whether a function is a hyperbola given an equation.
  • Find the focal axis, foci, vertices, semitransverse axis, semiconjugate axis, and asymptotes of a hyperbola given a graph.
  • Find the focal axis, foci, vertices, semitransverse axis, semiconjugate axis, and asymptotes of a hyperbola given an equation in standard form.
  • Find the eccentricity of a hyperbola and describe the effect on its shape.
  • Write the equation of a hyperbola given its transverse and conjugate axis’.
  • Convert the general equation of a hyperbola to standard form.
  • Write the equation of an ellipse that has been rotated by a given angle.
  • Identify the equation of a conic section that has been rotated.
  • Classify conic sections by their discriminants.

Suggested Activities

Students will be able to:

  • Parent function graph review
  • Identify the graph and form for constant, linear, quadratic, cubic, square root, cube root, exponential, logarithmic, rational, and step functions.
  • Determine the domain and range of constant, linear, quadratic, cubic, square root, cube root, exponential, logarithmic, rational, and step functions.
  • Determine the range of rational, polynomial, radical, and power functions graphically and algebraically.
  • Determine numerically or graphically whether a function has odd or even symmetry.
  • Determine algebraically whether a function has odd or even symmetry.
  • Identify whether a function is increasing, decreasing, or constant based on a table of numerical data.
  • Identify intervals on which a function is increasing or decreasing graphically.
  • Identify absolute extrema from the graph of a function.
  • Identify local extrema from the graph of a function.
  • Define and identify maximums and minimums as they occur between increasing and decreasing regions for all parent functions.
  • Define boundedness for a function.
  • Determine the boundedness of all parent functions.
  • Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
  • Identify the end behavior of a function.
  • Express end behavior using limit notation.
  • Differentiate between infinite limit and one sided limit notation.
  • Identify the horizontal and vertical asymptotes of a graph.
  • Express horizontal and vertical asymptote using limit notation.
  • Determine graphically whether a function is continuous or discontinuous at a point.
  • Classify the type of discontinuity graphically.
  • Identify and graph piecewise functions.
  • Analyze piecewise functions.
  • Evaluate piecewise functions given an input.
  • Determine graphically and algebraically whether a piecewise function is continuous or discontinuous at a point.
  • Define and determine the composition of two given functions
  • Find the domain of the composition of two given functions.
  • Decompose a given function as the composition of simpler functions.
  • Analyze a real-world situation of composition of functions.
  • Identify and verify implicitly two functions given a relation.
  • Determine if a function is one-to-one using the Horizontal Line Test.
  • Determine an inverse function graphically and algebraically. Graph an inverse by exchanging the coordinates of the points.
  • Verify that two functions are inverses of one another algebraically and graphically.
  • Compare domain and ranges of the function and its inverse.
  • Determine if the inverse is a relation or a function.
  • Restrict the domain of a function such that it will be invertible.
  • Recognize function transformations written in af(b(x-c))+d form.
  • Apply function transformations written in af(b(x-c))+d form to a given parent function.
  • Write the equation of a function under a horizontal or vertical reflection when described graphically or verbally.
  • Write the equation of a function under a vertical or horizontal stretch or shrink when described graphically or verbally.
  • Graph the equation of a function under multiple transformations.
  • Write the equation of a function under multiple transformations.

Suggested Activities

Students will be able to:

  • Compare graphs of power functions with differing powers.
  • Identify the power and constant of variation of the power function.
  • Determine graphically and algebraically which power functions are even and which are odd.
  • Relate the end behavior of a polynomial to the sign of the leading coefficient and its power - leading term test
  • Translate between verbal and algebraic descriptions of power functions.
  • Analyze key features of a power function: domain, range, continuity, symmetry, asymptotes, min/max, and end behavior.
  • Express end behavior using limit notation.
  • quadratics review/even degree
  • Sketch even powered polynomial functions with key components.
  • Identify potential number of roots for a quadratic and expand to higher powered even polynomials.
  • Apply the leading term test to determine the orientation of the graph and end behavior.
  • Express the end behavior of the polynomial using limit notation.
  • Identify regions of increase and decrease.
  • Identify local and absolute maximum and minimum values graphically.
  • Find the vertex and roots of a quadratic equation algebraically.
  • Solve a polynomial with imaginary roots using the quadratic formula.
  • Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
  • Identify and approximate irrational zeroes of a quadratic using a graphing calculator.
  • Identify the multiplicity of a factor.
  • Determine the effect of multiplicity on the behavior of a root.
  • cubics review/odd degree
  • Determine the minimum number of possible roots for an odd powered polynomial.
  • Sketch odd powered polynomial functions with key components.
  • Identify the potential number of roots for a cubic function and expand to higher powered odd polynomials.
  • Use Intermediate Value Theorem to locate potential roots given a table of numerical data.
  • Identify and approximate irrational zeroes of a quadratic using a graphing calculator.
  • Apply the leading term test to determine the orientation of the graph and end behavior.
  • Express the end behavior of the polynomial using limit notations.
  • Identify regions of increase and decrease.
  • Identify local and absolute maximum and minimum values graphically.
  • Find the roots of a cubic function by factoring by grouping.
  • Identify the multiplicity of a factor.
  • Determine the effect of multiplicity on the behavior of a root.
  • Generate a polynomial by converting roots to factors and multiplying.
  • Identify the multiplicity of a factor before expanding.
  • Determine a polynomial given its zeroes.
  • Expand a power of a binomial expression using the Binomial Theorem.
  • Generate the coefficients for a binomial expansion using Pascals triangle.
  • Generate the coefficients for a binomial expansion using the binomial coefficient formula.
  • Convert imaginary and irrational roots to factor form and multiply to generate a polynomial function with real coefficients.
  • Find the roots of a perfect cube binomials by using the formula for sums and differences of perfect cubes.
  • Identify the exact number of complex zeroes of a given polynomial.
  • Determine whether the complex conjugate of a zero is also a zero of a polynomial.
  • Find the quotient of two polynomials using long division.
  • Given a root, use long division to identify other roots for a polynomial.
  • Find the quotient of two polynomials using synthetic division.
  • Given a root, use synthetic division to identify other roots for a polynomial.
  • Use the Remainder Theorem to evaluate functions at a given value.
  • Identify all possible rational roots for a polynomial using the Rational Root Theorem (p/q).
  • Determine possible number of positive and negative roots using Descartes Rule of Signs.
  • Determine which possible rational roots are actual roots using synthetic division.
  • Express a polynomial as a product of linear and irreducible quadratic factors.
  • Solve a polynomial completely using long division, synthetic division, factoring and the quadratic formula to identify all roots.

Suggested Activities

Students will be able to:

  • Sketch the graph of a rational function as transformations of 1/x.
  • Identify vertical and horizontal asymptotes.
  • Explain the difference between a vertical and horizontal asymptote.
  • Identify any holes that occur.
  • Describe the end behavior of a rational function using limit notation.
  • Describe the left and right side behavior around any discontinuities using limit notation.
  • Determine the domain of a rational function.
  • Compose sums and differences of rational functions by finding a common denominator.
  • Analyze the graph of complex rational functions.
  • Identify the roots of a rational function.
  • Find any vertical asymptotes for a rational function.
  • Determine when a rational function will have no vertical asymptotes.
  • Identify rational functions that will have oblique asymptotes.
  • Find the equation of oblique asymptotes using long division.
  • Identify the domain and range of complex rational functions.
  • Describe the end behavior of complex rational functions using limit notation.
  • Sketch the graph of rational functions.
  • Solve a rational equation by clearing fractions.
  • Determine whether solutions are valid or extraneous.
  • Use rational functions to solve real world applications.

Suggested Activities

Students will be able to:

  • Determine the domain over which solutions to a polynomial are zero, positive and negative graphically.
  • Identify critical values for a polynomial.
  • Explain the effect even multiplicity has on a factor in a sign chart.
  • Determine the domain over which solutions to a polynomial are zero, positive and negative algebraically using a sign chart.
  • Sketch a graph for a polynomial inequality using a sign chart.
  • Solve a polynomial inequality using a sign chart.
  • Determine the domain over which solutions to a rational function are undefined, zero, positive and negative graphically.
  • Identify critical values for a rational function.
  • Determine the domain over which solutions to a rational function are undefined, zero, positive and negative algebraically using a sign chart.
  • Sketch a graph for a rational function inequality using a sign chart.
  • Solve a rational inequality using a sign chart.
  • Determine the domain over which solutions to a square root function are undefined, zero, positive and negative graphically.
  • Identify critical values for a square root function.
  • Determine the domain over which solutions to a square root function are undefined, zero, positive and negative algebraically using a sign chart.
  • Sketch a graph for a square root function inequality using a sign chart.
  • Solve a square root inequality using a sign chart.

Suggested Activities

Students will be able to:

  • Review Exponential Graphs and Equations
  • Identify and graph an exponential function using technology.
  • Identify and sketch the graph of an exponential function using a table of values.
  • Sketch the graph of an exponential function using transformations.
  • Identify key attributes of an exponential graph from an equation, including y-intercept, horizontal asymptote and end behavior.
  • Express end behavior of an exponential function using limit notation.
  • Evaluate an exponential function.
  • Differentiate between graphs and equations of exponential growth and decay.
  • Review Logarithmic Graphs and Equations
  • Identify and graph a logarithmic function using technology.
  • Explain the inverse relationship between logarithmic and exponential functions.
  • Identify key attributes of logarithmic graphs including roots, y-intercepts and asymptotes.
  • Express end behavior of logarithmic graphs using limit notation.
  • Evaluate a logarithmic expression.
  • Convert a logarithmic expression to an exponential expression.
  • Convert an exponential expression to a logarithmic expression.
  • Simplify and expand logarithmic expressions using properties of logarithms.
  • Define the natural logarithm.
  • Define the common logarithm.
  • Evaluate a logarithm by changing the base.
  • Simplify a natural logarithm using properties of logarithms and laws of exponents.
  • Find rates of growth and decay for exponential functions given real world applications.
  • Find the half life of a radioactive substance.
  • Model population growth using exponential functions.
  • Model a cooling problem using Newton’s Law of Cooling.
  • Identify and evaluate a logistic function.
  • Solve an exponential equation using a logarithm.
  • Solve a logarithmic equation.
  • Model economic growth and decay given real world applications.
  • Compute interest compounded annually, quarterly, and monthly.
  • Compute interest compounded continuously.
  • Compute annual percentage yield (APY).
  • Compute present value for an amount with a given interest rate.

Suggested Activities

Students will be able to:

  • Define a sequence explicitly.
  • Graph a sequence using technology.
  • Explain the difference between a sequence and a series.
  • Express the end behavior of a sequence using limit notation.
  • Generate a non-arithmetic, non-geometric sequence.
  • Determine whether a given sequence is arithmetic.
  • Define an arithmetic sequence explicitly.
  • Generate an arithmetic sequence.
  • Determine whether a sequence is geometric.
  • Define a geometric sequence explicitly.
  • Identify compound interest as a type of geometric sequence.
  • Generate a geometric sequence.
  • Define a partial sum of an infinite sequence.
  • Write out the sequence of partial sums for a series.
  • Find the nth partial sum of an arithmetic sequence.
  • Find the nth partial sum of a geometric sequence.
  • Express the nth partial sum of an arithmetic sequence using sigma notation.
  • Express the nth partial sum of a geometric sequence using sigma notation.
  • Evaluate a sum expressed in sigma notation.
  • Determine the limit of an infinite sequence from the sequence of partial sums.
  • Determine if an infinite sequence will converge or diverge.
  • Determine the conditions for a geometric sequence to converge.
  • Find the sum of an infinite geometric series if it exists.
  • Demonstrate mathematical induction by showing that substituting k+1 into a formula generates the same answer as substituting k and then adding on a k+1 term.

Suggested Activities