Analytic geometry creates a connection between graphs and equations. For example, the linear function f(x) = x2 – 2 (an equation) can also be represented by a graph:
Euclidean Geometry is based solely on geometric axioms without formulas or co-ordinates; Analytic geometry is the “marriage” of algebra and geometry with axes and co-ordinates [1].
Calculus and Analytic Geometry
Calculus and analytic geometry have become so intertwined, it’s rare nowadays to find a course in pure “Analytic Geometry”. It’s more common to take a course in Calculus and Analytic Geometry, which blends the principles of basic analytic geometry with concepts like functions, limits, continuity, derivatives, antiderivatives, and definite integrals.
Topics in Analytic Geometry A to Z
- Arc Length Formula: An “arc” is a curve segment; The arc length formula tells you how long this segment is.
- Area of a Bounded Region: are of a shape contained within a set of functions.
- Area under the curve: Calculating the area between a graph and the x-axis.
- Centroid: The average of all points in an object (e.g. the center of volume or mass).
- Center function: gives the trilinear coordinates of a triangle’s center.
- Distance Formula / Function: measures the distance between two points in a set (e.g. on a line).
- Delta x / Delta y: Distance traveled along the x- or y-axis.
- Displacement Function: gives us how far a particle has moved from a starting point at an given time.
- Distance Traveled (using derivatives).
- Intersection of lines: The place where two or more graphs cross each other.
- Length of a Line Segment: Measuring “how far” along an x or y axis.
- Parabola: a u-shaped curve; The graph of a quadratic function.
- Parallel Cross Sections: repeated cross sections for a solid, parallel to each other.
- Polar coordinates: “Circular” coordinates on a plane.
- Rate of change: a measure (a rate) of how things are changing.
- Slope: the ratio of a change in x (δx) to a change in y (δy).
- Quadrant: one of the four regions of the Cartesian plane / x-y axis.
- Riemann Sums: Estimating the area under a curve with rectangles.
- Secant line: A secant line connects two ore more points on a curve; An external secant is the “outside” part of the secant line.
- Sketching Graphs on the Cartesian Plane.
- Spherical coordinates: Coordinate system on a sphere.
- Tangent line: a line that touches a graph at only one point and is practically parallel. See also: Vertical Tangents and Horizontal Tangents.
- Tautochrone Problem / Brachistochrone: Classic problems about swinging pendulums.
- Testing for Symmetry of a Function.
- Transformations: shifts, dilations and other “movement” along the x or y axis.
- Vectors: show magnitude and direction.
- Velocity: Rate of change of displacement.
- x, y coordinate system: A system with a horizontal (x) axis and vertical (y) axis.
- x and y intercepts: The points where a graph crosses the x-axis or y-axis.
References
[1] Analytic Geometry and Calculus. Retrieved May 3, 2021 from: math.uci.edu/~ndonalds/math184/analytic.pdf
Stephanie Glen. "Analytic Geometry" From CalculusHowTo.com: Calculus for the rest of us! /problem-solving/analytic-geometry/
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