Checklist of Skills for Subtest I

If you found this site useful, be an evangelist and please spread the word to other CSET Single Subject Math examinees, and support this site with a contribution via Paypal!

What are the checklist of skills one must possess for Subtest I?

BASIC SKILLS: 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.

For COMPOSITE 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. a^x or e^x) functions.
• Know the relationship between logarithmic (log₁₀ x or log _ex) and exponential (a^x or e^x) 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 VECTORS you 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.

Hankering for more CSET Math Qs? Click on Need More Practice Questions? To purchase a vast database of Qs that I've compiled from my experience as a test taker and CSET Single Subject Math instructor at Cal State, San Bernardino, email me at innovationguy@yahoo.com. The CSET Qs on this site are taken from that collection of questions!

If you found this site useful, be an evangelist and please spread the word to other CSET Single Subject Math examinees, and support this site with a contribution via Paypal!