CSETMathGuru: THE Site for Single Subject Math

What skills must one possess for Subtest I: Algebra?

You must:

• Know distance formula.
  
  
• Midpoint formula
  
  
• Know symmetricity of graphs with respect to x-axis, y-axis, origin, y = x, line y = -x.
  
  
• Know the points slope form and slope intercept form of equation of lines.
  
  
• Know the condition for parallel and perpendicular lines using slopes.
  
  
• Prove the theorem on parallel and perpendicular lines using slopes.
  
  
• Know the definition of a function and relation.
  
  
• Know the Vertical Line Test for determining if a set of points belongs to a function.
  
  
• Be able to determine the domain and range of a function algebraically and graphically.
  
  
• Be able to factorize expressions using (a ± b)2, (a ± b)3, a2 – b2 , a3 ± b3
• Know how to complete the square of a quadratic expression.
  
  
• Determine if a function is even/odd.
  
  
• Be able to rationalize the denominator to simplify complex (i) and irrational (√) expressions
  
  

For QUADRATIC FUNCTIONS, you must:

• Be able to derive the quadratic formula
  
  
• Derive the sum of roots of a quadratic equation is –b/a and the product of roots is c/a.
  
  
• Know the relationship between the roots of a quadratic equation and its graph based on the discriminant, D = b2 – 4ac, being positive, negative or equal to zero
  
  
• Be able to convert a quadratic function in standard form to parabola form by completing the square.
  
  
• Determine the vertex, axis of symmetry, roots/x-intercepts of a quadratic function/ parabola.
  
  
• Determine if a parabola opens up or down, and based on that, know if the function has a maximum or minimum.
  
  
• Be able to solve variations of quadratic equations using substitution: for example, ax4 + bx2 + c = 0, √(ax4 + bx2 + c) = 0, ax2 + b/x2 + c = 0, √(ax + b) + √ (cx + d) = 0, etc. and find the maximum/ minimum of such functions.
  
  
• Be able to derive the coordinates of the vertex of a quadratic function in standard form.
  
  
• Model word problems related to quadratic functions and determine maximum/ minimum values.
  
  
• Determine roots/ x intercepts through quadratic formula/ completing the square.
  
  
• Know properties of complex numbers: addition, subtraction, multiplication, and division.
  
  
• Be able to represent the sum, difference, product and quotient of complex numbers in a + bi form.
  
  
• Know that irrational and complex roots of functions occur as conjugates.
  
  
• Be able to graph and interpret inequalities relating to quadratic functions: know for which intervals the graph lies above/ below the x-axis
  
  

For POLYNOMIAL FUNCTIONS, you must:

• Be able to determine the number of turning points for a function.
  
  
• Determine the end behavior of the function.
  
  
• Determine by visual inspection if a graph represents a certain polynomial function.
  
  
• Know characteristics of even/ odd functions.
  
  
• Know shapes of the most important BASE GRAPHS:

1. f(x) = √x
   
   
2. f(x) = x2
   
   
3. f(x) = xn, for n = odd/ even
   
   
4. f(x) = 1/x
   
   
5. f(x) = 1/xn, for n = odd/ even
   
   
6. f(x) = 3√x
   
   
7. f(x) = n√x, for n = odd/ even
   
   
8. f(x) = |x |
   
   

• Be able to transform the above base graphs for the following cases:
  
  

1. Given f(x), graphing f(x + c) or f(x – c)
   
   
2. Given f(x), graphing f(x) + c or f(x) – c
   
   
3. Given f(x), graphing c f(x), for c > 1 and c <>
   
   
4. Given f(x), graphing f(-x) and –f(x)
   
   

• Be able to graph polynomial functions in factored form
  
  

1. X-intercepts
   
   
2. Y-intercepts
   
   
3. End behavior of function
   
   
4. Intervals where graph is above/ below the x-axis by using a sign table.
   
   

• Be able to divide two polynomial functions using long division
  
  
• Be able to divide a polynomial and a linear (binomial) expression using Synthetic Division, and determine the Quotient and Remainder
  
  
• Know the proofs of Remainder Theorem and Factor Theorem
  
  
• Use the Factor Theorem and Remainder Theorem to determine if a monomial is a factor of the polynomial function.
  
  
• Be able to prove the Rational Roots Theorem.
  
  
• Apply the Rational Roots Theorem to determine the possible roots of a polynomial.
  
  
• Apply the Descartes Rule of Signs to determine the number of positive, negative and imaginary roots of a polynomial.
  
  
• Determine the polynomial function given its roots and their multiplicity
  
  
• Be able to calculate the roots, real and imaginary, of a polynomial.
  
  
• Be able to graph and interpret inequalities relating to polynomial functions: know for which intervals the graph lies above/ below the x-axis.
  
  

For RATIONAL FUNCTIONS, f(x) = P(x)/ Q(x), you must:

• Be able to graph various forms of rational functions by determining its:
  
  

1. x intercepts (solve: numerator = 0)
   
   
2. y intercepts (substitute: x = 0 into the function)
   
   
3. vertical asymptotes (solve: denominator = 0)
   
   
4. horizontal asymptotes (imagine x to be a large number, M, and simplify.)
   
   
5. slant asymptotes, if any, if the numerator has degree 1 more than the denominator
   
   

For INVERSE OF FUNCTIONS you must

• Be able to determine the inverse of functions (swap x and y in the original function and solve for y)
  
  
• Know properties of inverse functions
  
  
• Know the behavior of the inverse function, when given the graph of the original function (ie. the inverse function and the original function are symmetrical about the line y =x).
  
  
• Know the relationship of the domain and range of inverse functions and the original function.
  
  
• Know the horizontal line test for determining if a function has an inverse.
  
  
• Know what a one-to-one function is.
  
  

ForCOMPOSITE FUNCTIONS,f(x) = h(g(x)); h(x)/g(x); h(x)g(x), etc. you must

• Be able to determine the domain and range.
  
  
• Find values of composite functions for given values of x (x = a…)
  
  

For LOGARITHMIC AND EXPONENTIAL FUNCTIONS you must

• Know the general form (algebraically and graphically) of logarithmic (ie. log x) and exponential (ie. ax or ex) functions.
  
  
• Know the relationship between logarithmic (log10 x or log ex) and exponential (axor ex) functions, and transform exponential to logarithmic functions and vice versa.
  
  
• Be able to calculate the inverse of logarithmic and exponential functions.
  
  
• Graph logarithmic and exponential functions using elementary transformations
  
  
• Be able to determine the domain and range of transformed logarithmic and exponential functions.
  
  
• Know the properties of logarithms and exponents, and apply properties to simplify expressions
  
  
• Solve equations and inequalities with logs and exponents using definitions and properties of logs and exponents
  
  
• Be able to determine extraneous solutions to equations/inequalities
  
  
• Know the formula for compound interest for the case of continuous compounding
  
  
• Be familiar with exponential growth and decay situations, and related doubling time and half-life problems
  
  

For VECTORSyou must:

• Be able to determine the magnitude and direction of a vector
  
  
• Know the representation of vectors in standard (ai + bj) and component form
  
  
• Know basic vector operations
  
  
• Be able to determine unit vectors in a given direction
  
  
• Be able to calculate the direction angles of vectors
  
  
• Apply elementary vector properties to solve real-world problems
  
  
• Be able to calculate the angle between 2 vectors using dot product of vectors, and determine if 2 vectors are parallel or perpendicular
  
  

For SYSTEMS OF EQUATIONS, you must:

• Be able to solve systems of equations (lines, circles, ellipses, parabolas, hyperbolas) simultaneously to determine points of intersection.
  
  
• Solve systems of inequalities and be able to graph the shaded region representing all possible values of (x, y).
  
  
• Be able to determine the maximum and minimum values of an objective function of a linear programming problem/ situation.
  
  
• Be able to solve word problems involving a linear programming situation: choose appropriate variables, determine constraints and objective function, plot lines representing constraints, determine corner points for shaded polygon and calculate maximum/ minimum values of the objective function.
  
  
• Be able to solve systems of linear equations by finding the reduced echelon form of a matrix by performing row transformations
  
  
• Be able to classify systems of linear equations as consistent/ inconsistent/ possessing infinite solutions.
  
  

For MATRICES and DETERMINANTS, you must:

• Know basic properties of matrices
  
  
• Know the criteria for multiplying matrices
  
  
• Be able to multiply matrices
  
  
• Know properties of matrix multiplication.
  
  
• Be able to find the inverse of a matrix using determinants.
  
  
• Be able to calculate the inverse of a matrix by reducing it to echelon form.
  
  
• Be able to solve systems of linear equations by calculating the inverse of a matrix.
  
  
• Be able to find the value of a 2 X 2 and 3 X 3 determinants.
  
  
• Know properties of determinants pertaining to row and column transformations.
  
  
• Know Cramer’s Rule for solving linear equations using determinants.
  
  

For SEQUENCES, SERIES, PERMUTATIONS AND COMBINATIONS you must:

• Be able to determine the general term of an arithmetic/ geometric series using the common difference/ ratio.
  
  
• Be able to calculate the sum of n terms of an arithmetic/ geometric series.
  
  
• Be able to calculate the sum of an infinite geometric series.
  
  
• Know the principle of Mathematical Induction.
  
  
• Be able to apply the Principle of Mathematic Induction to prove elementary propositions/ conjectures.
  
  
• Know the (r + 1)th term in the expansion of (a + b)n using the binomial theorem.
  
  
• Expand binomial expressions using the binomial theorem and calculate specific terms.
  
  
• Be familiar with Pascal’s Triangle.
  
  
• Know the fundamental counting principle.
  
  
• Be able to calculate the number of different permutations & combinations of r elements that can be obtained from a set of n elements.