write-up #0 Assignment 0 **Introduction**

## The graphs of f(x),f(y),f(x+n),f(y+n)...

In the following, where you examine multiple graphs, consider
the different impact and observations you can make by a sequence
of graphs, adding one graph

at a time, in contrast to creating all graphs in a set at once.

Try to draw some graph by Graphic Calculator!

(Experiment 1) The equations and their graghs,

(Experiment 2) The equations and their graghs,

(Experiment 3) The equations and their graghs,

The graph passes the common point of (2,0).
(Experiment 4) The equations and their graghs,

(Experiment 5) The equations and their graghs,

(Experiment 6) The equations and their graghs,

Replace the 5 in each side of the equation with other small
values. It is more interesting if you keep multiple graphs on
the same axes.

1) n=1 (green), 2 (light blue), 3
(blue), 4 (violet), 5 (red)

2) n=1 (light blue), 2 (green), 3
(violet), 4 (blue), 5 (red)

3) n=1 (light blue), 2 (green), 3
(blue), 4 (violet), 5 (red)

(Experiment 7)

There is the equation of .The gragh is here.

Now.Let's play with this graph!

**1) Replace "5" in each side of the equation**

The "5" of the left hand replace other values. As
the values increase(n>0), the curves in 2nd and 3rd quadrant
is leaving from y-axis, as follows, but the curve passes the common
points of (0,0),(5,0).

The 5 of the right hand replace other values . When the values
change(n>0),the graphpasses the two point(0,0)and (the value
of change,0).

Next, replace "5" in both side!

**2) Replace "x" by "x-n" in each side
of the equation**

Replace x replace by x-2, the shape of the graph doesn't change,
but the position of the graph slips 2 to positive direction of
x-axis. When x is replaced by x-n, the position of the graph slips
n to positive dorection of x-axis. The equation and the graph
are here.

(red) n=10......

(purple) n=2...

(blue) n=-7.....

**Explanation**

I investigated the equations and thier graghs,considering the
coefficient of x,y as a variable number.Here,I look at them again,
particularly quadratic equations ,paying attention to the common
things of the 7 experiments.

1. Replacing x by x-n, the gragh of the equation was moved
"n" toward positive x axis. In the same way,replacing
y by y-n, the gragh of the equation was moved "n" toward
positive y axis.

What any kinds of graghs does it happen ?

2.Replacing x by nx, the slope of gragh changed.

3.Then , if the coefficient of xy is replaced by other values,
what happens? New problem happened.

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