TSIA2 (Texas Success Initiative)

Algebraic reasoning is the largest component of TSI Mathematics. This domain covers expressions, equations, inequalities, and functions.

An algebraic expression contains variables, constants, and operations. Terms are separated by addition or subtraction. Like terms have identical variable parts and can be combined.

Combine like terms by adding or subtracting their coefficients. Apply the distributive property to remove parentheses: a(b + c) = ab + ac.

To evaluate an expression, substitute given values for variables and compute following order of operations.

Common translations:

"Five more than x" = x + 5

"Three less than y" = y − 3

"Twice a number" = 2n

"The quotient of a and b" = a/b

"A number squared" = n²

A linear equation has variables raised only to the first power. The standard form is Ax + By = C. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

Isolate the variable by performing inverse operations on both sides of the equation. Whatever operation you perform on one side, perform on the other to maintain equality.

Steps:

Simplify each side (distribute, combine like terms)

Move variable terms to one side using addition or subtraction

Move constant terms to the other side

Divide both sides by the coefficient of the variable

Systems of linear equations can be solved by substitution or elimination.

Substitution method: Solve one equation for one variable, then substitute that expression into the other equation.

Elimination method: Multiply equations by constants so that adding the equations eliminates one variable. Solve for the remaining variable, then substitute back.

A system has one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).

Compound inequalities combine two inequalities. "And" means both must be true (intersection); "or" means at least one must be true (union).

Slope measures the steepness and direction of a line. Calculate slope using two points:

m = (y₂ − y₁) / (x₂ − x₁)

Positive slope: line rises from left to right. Negative slope: line falls from left to right. Zero slope: horizontal line. Undefined slope: vertical line.

From slope-intercept form (y = mx + b):

Plot the y-intercept (0, b)

Use slope to find additional points (rise over run)

Draw the line through the points

From standard form (Ax + By = C):

Find x-intercept by setting y = 0

Find y-intercept by setting x = 0

Plot both intercepts and draw the line

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals (their product equals −1).

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0.

Factor the quadratic expression, set each factor equal to zero, and solve. This method works when the quadratic factors nicely.

Common factoring patterns:

x = (−b ± √(b² − 4ac)) / 2a

The discriminant (b² − 4ac) determines the nature of solutions:

The graph of a quadratic function is a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward.

Vertex form: y = a(x − h)² + k, where (h, k) is the vertex.

The axis of symmetry passes through the vertex: x = h, or x = −b/(2a) in standard form.

Product rule: aᵐ × aⁿ = aᵐ⁺ⁿ

Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ

Power rule: (aᵐ)ⁿ = aᵐⁿ

Zero exponent: a⁰ = 1 (a ≠ 0)

Negative exponent: a⁻ⁿ = 1/aⁿ

Fractional exponent: a^(m/n) = ⁿ√(aᵐ)

Factor out perfect squares from under the radical. √(ab) = √a × √b. Rationalize denominators by multiplying by appropriate forms of 1.

Add or subtract radicals only when they have the same radicand (number under the radical). Multiply radicals by multiplying the radicands: √a × √b = √(ab).

A polynomial is an expression with one or more terms, each consisting of a coefficient and variables raised to non-negative integer powers.

Combine like terms (terms with identical variable parts).

Distribute each term of one polynomial to every term of the other. For binomials, use FOIL: First, Outer, Inner, Last.

Use long division or synthetic division (for divisors of the form x − c).

A function assigns exactly one output to each input. Function notation f(x) represents the output when x is the input.

To find f(3), substitute 3 for x in the function definition and simplify.

Domain: the set of all possible input values. Range: the set of all possible output values.

For polynomial functions, the domain is typically all real numbers. For rational functions, exclude values that make the denominator zero. For square root functions, the radicand must be non-negative.

The vertical line test determines if a graph represents a function: if any vertical line intersects the graph more than once, it is not a function.

Identify intercepts, maximum and minimum values, and intervals of increase and decrease.