Math III Exam Preparation Notes

Math III Notes

Unit 1: Introduction to Functions

• Relation: A set of ordered pairs.
• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Vertical Line Test: A method to determine if a graph represents a function; if a vertical line intersects the graph at more than one point, it is not a function.

Finding Domain and Range

• Example 1: If given $f(x) = x^2$, the domain is all real numbers and the range is $y \geq 0$.
• Example 2: If given points like {(−3,−5),(−1,1),(0,4),(1,7)}, domain is {−3,−1,0,1} and range is {−5,1,4,7}.

Evaluating Functions

• If $f(x) = 2x + 5$, to find $f(3)$, substitute 3 for x: $f(3) = 2(3) + 5 = 11$.

Function Rules

• Write a function for monthly costs, e.g., for cable services. ( C(x) = 24.50 + 3x ) where x is the number of channels.

Unit 2: Exponentials & Logarithms

Exponential Growth

• The general form of an exponential function is $y = a(1 + r)^t$, where a is the initial amount, r is the growth rate, and t is time.

Logarithmic Form

• The formula $y = a^x$ can be rewritten as $x = ext{log}_ay$.

Finding Areas and Circumference

• The area of a circle: $A = \pi r^2$; the circumference: $C = 2 \pi r$.

Unit 3: Polynomials

Factoring

• GCF: Factor out the greatest common factor.
• Difference of Squares: ( a^2 - b^2 = (a-b)(a+b) ).
• Trinomials: For example, how to factor ( x^2 + 5x + 6 ) as ( (x+2)(x+3) ).

Solving Quadratics

• Use factoring, completing the square, or the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Unit 4: Rational Functions

Definition

• A rational function is a ratio of two polynomials:
  $f(x) = \frac{p(x)}{q(x)}$.

Asymptotes

• Vertical Asymptotes: Found by setting the denominator equal to zero.
• Horizontal Asymptotes: Determine by the degrees of the numerator and denominator.

Unit 5: Geometry

Triangle Properties

• Midsegment: A segment connecting the midpoints of two sides of a triangle.
• The midsegment is parallel to the third side and half its length.

Congruent Triangles

• SSS, SAS, ASA, AAS, HL postulates are used to prove triangle congruence.

Parallelograms

• Properties include opposite sides are equal, opposite angles are equal, and the diagonals bisect each other.

Unit 6: Circles

Circle Parts

• The radius of a circle is defined as the distance from the center to any point on the circle.
• The diameter is twice the radius.

Areas & Circumferences

• Area: $A = \pi r^2$; Circumference: $C = 2 \pi r$.

Arcs

• Length of an arc: $L = \text{(central angle/360)} * C$.
• Area of a sector: $A = \frac{1}{2} r^2 \text{(central angle in radians)}$.

Unit 7: Volume and Surface Area

Formulas

• Volume for a cylinder: $V = \pi r^2 h$.
• Surface Area for a cube: $SA = 6s^2$.

Word Problems

• Apply geometric formulas to real-world contexts such as construction problems, volume of containers, and surface areas of various shapes.

Unit 8: Trigonometry

Basic Trig Ratios

• SOH-CAH-TOA: Definitions of sine, cosine, and tangent based on a right triangle.
• Use the ratio definitions to solve for angles and side lengths.

Unit Circle

• Important for sine and cosine values within trigonometric functions. The unit circle provides a geometric interpretation of the functions.

Equations and Functions

• To model periodic behaviors using equations of sine and cosine functions; includes amplitude, period, and phase shifts.