Precalculus

Upon completion of MATH 1110 the student should be able to:

FUNCTIONS

• understand the concept of function and be able to determine which relations are functions by an examination of the equation and/or the graph of the relation.
• find the domain of any function and the range of functions for which the inverse can be determined or for which the graph can be easily sketched.
• extract the functional rule from a 'word problem'.
• determine if a function is odd or even and understand the graphical implication of the property.
• sketch the graphs of the following functions:

            y = x, y = x², y = x³, y = |x|, y = sqrt(x), y = 1/x, y = 1/x², y = sqrt(a² - x²)

• sketch the graphs of the following transformations of the above functions:
  
  y = f(x) + c, y = f (x + c), y = -f(x), y = cf(x), y = f(cx)

• apply the above transformations to any given graph or function.
• sketch the graph of simple piece-wise defined functions.
• sketch the graph of any quadratic function and be able to determine all intercepts and the vertex using the quadratic formula and/or completing the square.
• determine the equation of a quadratic from its graphical properties.
• solve maximum-minimum 'word problems' involving a quadratic function.
• add, subtract, multiply and divide functions and be able to determine the domains of the resulting functions.
• determine the composite of several functions and its domain.
• determine the inverse of a given one-to-one function and the domain and range of the inverse function.
• prove that a given function is the inverse of another given function.
• sketch the graph of the inverse of a given one-to-one function when the inverse functional rule cannot be determined.

POLYNOMIAL AND RATIONAL FUNCTIONS

• find the quotient and remainder when a polynomial is divided by a second polynomial.
• use the remainder theorem.
• use the factor theorem to find the real roots of polynomial equations and the real zeros of polynomial functions.
• determine the multiplicity of zeros.
• use the rational root test to determine all possible rational roots.
• factor and graph any polynomial of degree n provided that the polynomial has at least n-2 rational roots.
• obtain the functional rule for a polynomial when given certain information about the roots and a value that satisfies the function and graph the function.
• sketch the graph of proper and improper rational functions that have a most one horizontal asymptote or an oblique asymptote.
• solve 'word problems' that involve polynomial or rational functions.

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

• find the exact value of logarithmic and exponential expressions.
• use a calculator to approximate the logarithm of a number to any base.
• use a calculator to approximate the solutions to exponential and logarithmic equations for all bases.
• find the inverse of a given exponential or logarithmic function and the domain and range of the inverse function.
• demonstrate an understanding of the rules of logarithms by rewriting given expressions.
• sketch the graph of exponential and logarithmic functions determining the value of all intercepts and the equation of the asymptote.
• solve 'word problems' which require the use of logarithms and/or exponentials; i.e. growth and decay problems and compound interest problems.

THE TRIGONOMETRIC FUNCTIONS

• convert radians to degrees, minutes and seconds and vice versa.
• solve problems that demonstrate an understanding of the relationship between the central angle, the arc length and the radius of a circle.
• solve problems that demonstrate an understanding of the relationship between the angular velocity, the linear velocity and the radius of a wheel or similar object.
• determine the area of a circular sector.
• demonstrate an understanding of the six trigonometric functions relative to a right triangle and to the unit circle.
• recall and apply the fundamental trigonometric identities, the co-function formulas and the formulas for negatives.
• sketch the graphs of the six basic trigonometric functions and recognize which functions are odd and which functions are even.
• determine amplitude, period, vertical shift and phase shift of any trigonometric function and sketch its graph showing all intercepts and turning points.
• find the exact values of the remaining trigonometric functions given the values of two trigonometric functions or the value of one trigonometric function and the quadrant.
• find the exact values of the trigonometric functions for an angle in standard position given a point on the terminal side.
• find the reference angle of any angle in degrees and/or radians.
• express any trigonometric function as a function of a given trigonometric function.
• recall the exact values of the trigonometric functions for reference angles of 30^o, 45^o, and 60^o and the axis angles.
• use a calculator to approximate the value of the trigonometric function of any real number.
• use a calculator to approximate the reference angle given the value of the trigonometric function.
• demonstrate an understanding of the terms 'angle of depression' and 'angle of elevation' and solve 'word problems' involving right triangles.

ANALYTIC TRIGONOMETRY AND APPLICATIONS

• recall or derive and demonstrate an understanding of the addition and subtraction formulas, the double angle formulas and the half-angle identities for sine, cosine and tangent.
• verify trigonometric identities.
• find all the solutions of trigonometric equations and find solutions on a restricted interval.
• sketch graphs of the six inverse trigonometric functions and state the domain and range of each function.
• find the exact value of inverse trigonometric expressions.
• simplify given composites of trigonometric and inverse trigonometric functions.
• solve 'word problems' that require the use of the inverse trigonometric functions.
• verify inverse trigonometric identities.

PARABOLAS, ELLIPSES AND HYPERBOLAS

• find the vertex, focus and directrix of a parabola and sketch its graph.
• find the vertices and foci of an ellipse and sketch its graph.
• find the vertices and equations of the asymptotes of a hyperbola and sketch its graph.
• find an equation of a parabola or ellipse that satisfies given conditions.
• write the equation of any conic section in standard form by completing the square.

OPTIONAL TOPICS

• solve 'word problems' that require the use of the Law of Sines and/or Law of Cosines.
• understand the polar coordinate system and be able to graph a function written in polar coordinates.
• sketch the graph of a plane curve given by a set of parametric equations.
• find parametric equations of basic plane curves.
• demonstrate an understanding of the product-to-sum and sum-to-product formulas.
• combine a sine function and a cosine function of the same period into a single cosine function.