# Page Comparison

## Key

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Next, decide on the type of regression. If you want to do a linear regression, the equation would take the form {\displaystyle

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body y=mx+b
}
. If you want a quadratic regression, the equation would look like {\displaystyle y=ax^{2}
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body --uriencoded--y=ax%5e%7B2%7D+bx+c
}
. The enterprising student can extrapolate this to any form they would like, such as {\displaystyle y=e^{m}x+bx}as
Mathinline
body --uriencoded--y=e%5e%7Bm%7Dx+bx
.

After you have decided the form your equation should take, substitute "{\displaystyle y}" with "{\displaystyle y_{1}}", "{\displaystyle x}" with "{\displaystyle x_{1}}", and "{\displaystyle =}" with {\displaystyle \sim }

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body y
" with "
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body --uriencoded--y_%7B1%7D
", "
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body x
" with "
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body --uriencoded--x_%7B1%7D
", and "
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body =
" with
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body \sim
. For example, to do a linear regression, you would enter {\displaystyle y_{1}
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body --uriencoded--y_%7B1%7D\sim mx_
{1}
 %7B1%7D+b
}
. This is saying: "find the constant values m and b that best satisfy {\displaystyle
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body y=mx+b
} for every {\displaystyle x} and {\displaystyle y}
for every
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body x
and
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body y
in my table, where {\displaystyle
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body --uriencoded--x=x_
{1}} and {\displaystyle
 %7B1%7D
and
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body --uriencoded--y=y_
{1}}"
 %7B1%7D
. See the examples below for a demonstration.

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It can be useful to restrict the domain or range of a function. To do this, add a restriction of the form {\displaystyle \left\{2<x<8\right\}} directly

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body --uriencoded--\left\%7B2
directly onto the end of a function. This works with any variable: for a function of {\displaystyle c} where c should be greater than 4, type {\displaystyle
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body --uriencoded--\left\
{c>4
 %7Bc>4\right\
}}
 %7D
.

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### Inequalities

Desmos also has the capability to graph inequalities. Simple inequalities are easy: type an expression followed by a comparison sign (e.g. {\displaystyle <}, {\displaystyle >=}

Mathinline
body <
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body >=
) and then a value. For example, {\displaystyle 3x<4}
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body 3x<4
. A slightly more complicated example might be {\displaystyle x^{2}>y>2}
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body --uriencoded--x%5e%7B2%7D>y>2

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You can also use this on functions you define. For example, say you're modeling compound interest, and want to set up an inequality for the space under the curve. To do this, define a function for amount given time, {\displaystyle

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body A(t)
}
, and an inequality {\displaystyle
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body 0
}
. Notice the use of {\displaystyle x} instead of {\displaystyle t}
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body x
Mathinline
body t
. Desmos is picky about variables, and for inequalities it implicitly defines
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body x
and
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body y
as the input and output.

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We can use this powerful tool in tandem with restrictions. To graph the area between functions {\displaystyle

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body f(x)
} and {\displaystyle
and
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body g(x)
}
, all we have to do is type {\displaystyle
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body --uriencoded--f\left(x\right)
{g
 %7Bg\left(x\right)>y>f\left(x\right)\right\
}}
 %7D
.

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## Built-in Functions and Symbols

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Input

Result

Explanation

theta

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body \theta

A variable, like

### Constants

Input

Result

Explanation

pi

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alignment center
\pi

The constant 3.14...

tau

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alignment center
\tau

The constant 6.28...

e

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Image Removed
 e

The constant 2.71...

### Exponent and Log Functions

Input

Result

Explanation

exp(x)

exp(x)

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body --uriencoded--e%5ex

ln(x)

ln(x)

The natural log of x

log(x)

log(x)

The log (base 10) of x

log_n(x)

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Mathblock
log_n x

The log (base n) of x

x^n

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alignment center
x^n

x to the nth power

sqrtx

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\sqrt{x}

The square root of x

nthrootx

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\sqrt[\leftroot{-3}\uproot{3}n]{x}

The generalized root function.

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### Trig Functions

sin(x)

arcsin(x) or

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body --uriencoded--\sin%5e%7B-1%7Dx

sinh(x)

cos(x)

arccos(x) or

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body --uriencoded--\cos%5e%7B-1%7Dx

cosh(x)

tan(x)

arctan(x) or

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body --uriencoded--\tan%5e%7B-1%7Dx

tanh(x)

sec(x)

arcsec(x) or

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body --uriencoded--\sec%5e%7B-1%7Dx

sech(x)

csc(x)

arccsc(x) or

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body --uriencoded--\csc%5e%7B-1%7Dx

csch(x)

cot(x)

arccot(x) or

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body --uriencoded--\cot%5e%7B-1%7Dx

coth(x)

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