an overview of analytic geometry

Analytic GeometryAn Overview of

Joseph Carlo Magsino Rosanna Brioso

2014

Table of Contents

Analytic GeometryAn Overview of 1

Points, LinesMidpointDistance between two pointsLinear EquationsFunctionsBasic FunctionsShifting, Scaling, ReflectionConic SectionsSources

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Points, LinesPOINTS are described by ordered pairs of numbers: the first of the two numbers tells how far to the right horizontally the point is from the origin (and negative means go left instead of right), and the second of the two numbers tells how far up from the origin the point is (and negative means go down instead of up). The horizontal coordinate is called the x-coordinate, and the vertical coordinate is called the y-coordinate.

The set of points are LINES. Remember: Slope of a line is rise over run, meaning vertical change divided by horizontal change (moving from left to right in the usual coordinate system).

The midpoint formula is used to find the coordinates of the midpoint of a line segment.

MidpointAnalytic GeometryAn Overview of 3Analytic GeometryAn Overview of 2

Distance between two points

The distance formula is used to find the distance between two points A(x1 ; y1) and B(x2 ; y2).

Linear Equations

Solution

General/Standard Form

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form.

Example Two-Point Form

Intercept Form

where (x1, y1) and (x2, y2) are two points on the line with x2 ≠ x1. This is equivalent to the point-slope form above, where the slope is explicitly given as (y2 − y1)/(x2 − x1).

where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form is in standard form with A/C = 1/a and B/C = 1/b.

Analytic GeometryAn Overview of 5Analytic GeometryAn Overview of 4

Slope-Intercept Form

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form.

Solution

Example

Point-Slope Form

where m is the slope of the line and (x1,y1) is any point on the line. The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y1) is proportional to the difference in the x coordinate (that is, x − x1).

Solution

ExampleFind the equation of the line that passes through

(−2,1) with slope of −3.


FunctionsA function relates an input to an output.

Basic Functions

It is like a machine that has an input and an output. And the output is related somehow to the input.💡

"...each element..." means that every element in X is related to some element in Y. We say that the function covers X (relates every element of it). (But some elements of Y might not be related to at all, which is fine.)

Two Important Things!💡

"...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input.

"

#


Shifting, Scaling, ReflectionShifting

- If c is a positive real number, then the graph of y=f(x) is shifted to the right by c units - If c is a negative real number, then the graph of y=f(x) is shifted to the left by c units

Vertical Shift

a translation in which the size and shape of a graph of a function is jot changed but the graph is.➡️

Horizontal Shift

- If c is a positive real number, the graph of f(x)+c is the graph of y=f(x) shifted upwards by c units. - If c is a positive real number, the graph of f(x)-c, the graph of y=f(x) shifted downwards by c units.


Examples

The function comes from the basic function f (x) = x with the constant 6 subtracted on the inside. This gives the basic function a horizontal shift RIGHT 6 units. So, take the basic graph of f (x) = x and shift RIGHT 6 units.

This function comes from the basic function f(x) = |x| with the constant 3 subtracted on the outside. This gives the basic function a vertical shift DOWN 3 units. So, take the basic graph of f(x) = |x| and shift it DOWN 3 units.

f(x)=|x|-3

f(x)= √x — 6

Reflection

The graph of the function y=f(-x) is the graph of y=f(x) reflected across the y-axis.

⬇️The graph of the function

y=f(x) is the graph of y=f(x) reflected across the x-axis.

⬇️

Vertical Stretching and Shrinking

If c>1, then the graph of y=cf(x) is the graph of y=f(x)

stretched vertically by c.

If 0<c<1, then the graph of y=cf(x) is the graph of y=f(x)

shrinks vertically by c.

⬇️ ⬇️

Horizontal Stretching and Shrinking

If c>1, then the graph of y=cf(x) is the graph of y=f(x)

shrunk horizontally by c.

If 0<c<1, then the graph of y=cf(x) is the graph of y=f(x) stretched horizontally by c.

⬇️ ⬇️


Examples

f(x) = -x3

This is a transformation on the basic function f(x) = x3. Since the negative is on the outside of the function, there is a reflection about the x-axis.

f(x)= √-x + 2

Horizontal Dilation

If x is replaced by x/a in a formula and a>1, then the effect on the graph is to expand it by a

factor of a in the x-direction (away from the y-axis).

⬇️If 0<a<1, then the effect on the graph is to contract by a factor

of 1/a (towards the y-axis).

⬇️

Vertical Dilation

If y is replaced by y/B in a formula and B>0, then the effect on the graph is to dilate it by a factor of B in the vertical direction. As before, this is an expansion or contraction depending on whether B is larger or smaller than one. Note that if we have a function y=f(x), replacing y by y/B is equivalent to multiplying the function on the right by B: y=Bf(x). The effect on the graph is to expand the picture away from the x-axis by a factor of B if B>1, to contract it toward the x-axis by a factor of 1/B if 0<B<1, and to dilate by |B| and then flip about the x-axis if B is negative.

➡️

We use the word "dilate'' to mean expand or contract.

💡

This function is a transformation on the basic function f (x) = x . Since we have a negative on the inside of the radical that gives us a reflection about the y-axis. We also have “+2” on the outside of the function, so this gives us a shift UP 2 units.


Conic SectionsDid you know that by taking different slices through a cone you can

create a circle, an ellipse, a parabola or a hyperbola?

Parts of a Cone

Axis - central line about which the cone is symmetric

Generator - line which when it rotates about the axis, sweeps the cone

A conic section is the i nt e r s e c t i o n o f a plane and a cone.

CircleThe cone is cut at right-angles to its axis.

Parts of a Circle

Center - fixed point

Radius - a line that joins the center of a circle with any point on the circumference of the circle

Diameter - a straight line segment that passes through the center of a circle


ParabolaThe cone is cut parallel to a generator.

Parts of a ParabolaThe Axis of Symmetry goes through the focus, at right angles to the directrix.

A parabola is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line called a directrix.

An equation of the parabola with focus (0,p) and directrix y=-p is

The Vertex is where the parabola makes its sharpest turn and is halfway between the focus and directrix.

Case 1: if p > 0 Case 2: if p < 0

Case 1: if p > 0 Case 2: if p < 0

EllipseThe cone is cut at an oblique angle shallower than the generator.

Parts of an Ellipse

The set of points in a plane where the sum of whose distances from two fixed points F1 & F2 is constant.

An equation of the parabola with focus (p,0) and directrix x=-p isAnalytic GeometryAn Overview of 19Analytic GeometryAn Overview of 18

An equation of an ellipse with foci (± c,0) and vertices

(± a,0) is

An equation of an ellipse with foci (0, ± c) and

vertices (0, ± a) is

HyperbolaA double cone is cut at an angle steeper than the generator. A hyperbola is the set of points in a plane the difference of whose distances from two fixed points F1 & F2 is constant.

Parts of a Hyperbola

An equation of a hyperbola with foci (± c,0), vertices (± a,0), and

asymptotes y= ±(b/a)x is

An equation of a hyperbola with foci (0, ± c), vertices (0, ± a), and

asymptotes y= ±(a/b)x is


Take any lamp with a cylindrical shade and observe the light cone's projection on the wall!

Put it next to the wall (hyperbola). Tilt (parabola).

Tilt more (ellipse). Align horizontal (circle).

Try This ✏️ Sources 🌎 🌏 🌎

http://pc30swinter2011.pbworks.com/w/page/36696016/Conics http://academic.sun.ac.za/mathed/shoma/Index14.htm http://www.uzinggo.com/ratio-circles-circumference-diameter/understanding-rational-irrational-numbers/algebra-foundations-grade-8 http://www.prof-desk.com/materials/conic-sections-part-1-745/?m=1 http://commons.wikimedia.org/wiki/File:Parts_of_a_Parabola.JPG http://sccs-tech.weebly.com/q6-ellipses.html http://coefs.uncc.edu/nabyars/index-of-nabyarsanalytic-geometry/new-page-4/ http://www.formyschoolstuff.com/school/math/glossary/H.htm http://coefs.uncc.edu/nabyars/index-of-nabyarsanalytic-geometry/new-page-5/ https://sites.google.com/site/mathclc/graphs http://www.uh.edu/engines/epi2556.htm http://www.mathsisfun.com/sets/function.html


an overview of analytic geometry

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