Analytic Geometry

Distance and Midpoint Formulas

Analytic geometry provides tools for calculating distances and midpoints between points in the coordinate plane.

• Distance Formula: The distance between points and is given by:

• Midpoint Formula: The midpoint of the segment joining and is:

• Example: For and : Distance: Midpoint:

Equations of Circles

The equation of a circle describes all points at a fixed distance (radius) from a center point.

• Standard Form:

• General Form:

• Example: Center , radius $10(x + 6)^2 + (y - 8)^2 = 100x^2 + y^2 + 12x - 16y + 0 = 100$ (expand and rearrange)

• Intercepts: Set or and solve for the other variable.

• Tangent Line: A tangent to the circle at a point touches the circle at exactly one point. If tangent to the y-axis, the radius equals the x-coordinate of the center.

Functions and Their Properties

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Example: For , the domain excludes (where denominator is zero): Domain:

Function Transformations

Transformations include shifting, stretching, compressing, and reflecting graphs.

• Vertical Shift: shifts up/down by units.

• Horizontal Shift: shifts left/right by units.

• Reflection: reflects over the x-axis; reflects over the y-axis.

• Stretch/Compression: stretches vertically by ; compresses horizontally by .

• Example: is a right shift of by 2 units.

Even and Odd Functions

A function is even if for all in the domain, and odd if .

• Example: is neither even nor odd.

One-to-One Functions and Inverses

A function is one-to-one if each output is produced by exactly one input. The horizontal line test is used to determine this from a graph.

• Inverse Function: If is one-to-one, its inverse satisfies .

• Example: For , .

Quadratic Functions

Vertex, Axis of Symmetry, and Concavity

The graph of a quadratic function is a parabola.

• Vertex:

• Axis of Symmetry:

• Concavity: If , parabola opens up (concave up); if , opens down (concave down).

• Example: Vertex: Axis: Concave up (since )

Intercepts

• Y-intercept: Set ;

• X-intercepts: Solve using the quadratic formula:

Domain, Range, and Intervals of Increase/Decrease

• Domain: for all quadratics

• Range: if concave up; if concave down, where is the vertex y-value

• Increasing/Decreasing: Decreasing on , increasing on , where is the vertex x-value

Maximum and Minimum Values

• Maximum: If , vertex is maximum

• Minimum: If , vertex is minimum

Quadratic Inequalities

• Solve or by finding roots and testing intervals.

Polynomial Functions

Definition and Degree

A polynomial function is an expression of the form where is a non-negative integer.

• Degree: The highest power of with a nonzero coefficient.

• Leading Term: The term with the highest degree.

• Constant Term: The term without .

Zeros and Multiplicity

• Zero: Value where

• Multiplicity: Number of times a zero is repeated

• Graph Behavior: If multiplicity is odd, graph crosses x-axis; if even, graph touches x-axis.

End Behavior and Turning Points

• End Behavior: Determined by leading term

• Maximum Number of Turning Points: for degree

Polynomial Construction

• Given zeros and degree, construct polynomial by multiplying factors for each zero .

• Example: Zeros (multiplicity 1), (multiplicity 2), degree 3:

Remainder and Factor Theorems

• Remainder Theorem: The remainder of divided by is .

• Factor Theorem: is a factor of if and only if .

Rational Zeros Theorem

• Possible rational zeros are factors of constant term over factors of leading coefficient.

Rational Functions

Domain and Asymptotes

• Domain: All real numbers except where denominator is zero.

• Vertical Asymptotes: Values of where denominator is zero and numerator is nonzero.

• Horizontal Asymptotes: Determined by degrees of numerator and denominator:

  Degree (Numerator)	Degree (Denominator)	Horizontal Asymptote
  Less than	Greater than
  Equal	Equal
  Greater than	Less than	None (may have oblique)

• Oblique Asymptotes: Occur when degree of numerator is exactly one more than denominator; found by polynomial division.

Exponential and Logarithmic Functions

Exponential Functions

• Form:

• Domain:

• Range: for

• Horizontal Asymptote:

Logarithmic Functions

• Form:

• Domain:

• Range:

• Vertical Asymptote:

• Properties:

Solving Exponential and Logarithmic Equations

• Isolate the exponential or logarithmic term, then use logarithms or exponentials to solve.

• Example: Combine: So

Applications: Exponential Decay

• Decay Model: , where

• Half-life:

• Example: For strontium-90,

Summary Table: Key Formulas

Additional info:

• Some questions reference graphing tools and graphical analysis; students should practice sketching graphs and identifying key features such as intercepts, asymptotes, and intervals of increase/decrease.

• For inequalities, always test intervals between roots to determine where the inequality holds.

• For polynomial construction, ensure the degree matches the sum of multiplicities.