## 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 + b*form.**i** - 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

**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:**

- f(x) = √x
- f(x) = x2
- f(x) = xn, for n = odd/ even
- f(x) = 1/x
- f(x) = 1/xn, for n = odd/ even
- f(x) = 3√x
- f(x) = n√x, for n = odd/ even
- f(x) = |x |

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

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

- Be able to graph polynomial functions in factored form

- X-intercepts
- Y-intercepts
- End behavior of function
- 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.

__, you must:__

**RATIONAL FUNCTIONS, f(x) = P(x)/ Q(x)**- Be able to graph various forms of rational functions by determining its:

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

__you must__

**INVERSE OF FUNCTIONS**- 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.

**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*…)

__you must__

**LOGARITHMIC AND EXPONENTIAL FUNCTIONS**- 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

__you must:__

**VECTORS**- Be able to determine the magnitude and direction of a vector
- Know the representation of vectors in standard (a
**i +**b**j)**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

__you must:__

**SYSTEMS OF EQUATIONS,**- 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.

__you must:__

**MATRICES and DETERMINANTS,**- 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.

__you must:__

**SEQUENCES, SERIES, PERMUTATIONS AND COMBINATIONS**- 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.