How to find the function of a graph calculator

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Number Line
What's this about?
How to reconstruct a function?
How to find a function through given points?
How to find a function with a given inflection point?
And how to use that in my example?
Can you graph functions on TI 84?
How do you calculate functions on a TI 84?

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Examples

line\:(-2,\:4),\:(1,\:2)
slope\:3x+3y-6=0
parallel\:2x-3y=9,\:(4,-1)
perpendicular\:y=4x+6,\:(-8,-26)
domain\:y=\frac{x^2+x+1}{x}
range\:y=\frac{x^2+x+1}{x}
asymptotes\:y=\frac{x}{x^2-6x+8}
extreme\:points\:y=\frac{x^2+x+1}{x}
intercepts\:f(x)=\sqrt{x+3}
f(x)=2x+3,\:g(x)=-x^2+5,\:f\circ \:g

functions-line-calculator

What's this about?

This is, in some way, the opposite of curve sketching. Curve sketching means you got a function and are looking for roots, turning and inflection points. What we do here is the opposite: Your got some roots, inflection points, turning points etc. and are looking for a function having those.

How to reconstruct a function?

Primarily, you have to find equations and solve them. This gives you the coefficients of your function. Here is an example: Let's assume we are looking for a function of degree having a minimum turning point at (1|-4) and a maximum turning point at (-1|3).

You are looking for a function with:
quadratic function
Maximum turning point at (-1|3)
Minimum turning point at (1|-4)

Mathepower found the following function:

This is the graph of your function.

Dein Browser unterst�tzt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P

Roots at -0.386; 3.886
y-axis intercept at (0|-1.5)
Maximum and minimum turning points at (1.75|-4.563)
Inflection points

This is how Mathepower calculated:

The point at (-1|3) gives the equation :

simplified: :
1a-1b+1c=3

The point at (1|-4) gives the equation :

simplified: :
1a+1b+1c=-4

So, we got the following system of equations: :

a	-1b	+c	=	3
a	+b	+c	=	-4

This is how to solve this system of equations:

a	-1b	+c	=	3
a	+b	+c	=	-4

a	-1b	+c	=	3
	2b		=	-7

( -1 times line 1 was added to line 2 )

a	-1b	+c	=	3
	b		=	-3,5

( The 2 line was divided by 2 )

2 line:	b+0c = -3,5
c can be chosen freely
Solve for b : :	b = 0c -3,5

1 line:
Substitute variables already known:
Solve for a :	a = -1c -0,5

Set a equal to
This means that c is equal to -1,5

Inserting shows that the function equals to ist.

How to find a function through given points?

The general rule is that for any n given points there is a function of degree whose graph goes through them. So e.g. you find by solving equations a function of degree through the four points (-1|3), (0|2), (1|1) und (2|4):

You are looking for a function with:
Function of degree 3
Point at (-1|3)
Point at (0|2)
Point at (1|1)
Point at (2|4)

Mathepower found the following function:

This is the graph of your function.

Dein Browser unterst�tzt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P

Roots at -2
y-axis intercept at (0|2)
Maximum and minimum turning points at (-0.913|3.014); (0.913|0.986)
Inflection points at (0|2)

This is how Mathepower calculated:

The point at (-1|3) gives the equation :

simplified: :
-1a+1b-1c+1d=3

The point at (0|2) gives the equation :

simplified: :
0a+0b+0c+1d=2

The point at (1|1) gives the equation :

simplified: :
1a+1b+1c+1d=1

The point at (2|4) gives the equation :

simplified: :
8a+4b+2c+1d=4

So, we got the following system of equations: :

-1a	+b	-1c	+d	=	3
			d	=	2
a	+b	+c	+d	=	1
8a	+4b	+2c	+d	=	4

This is how to solve this system of equations:

-1a	+b	-1c	+d	=	3
			d	=	2
a	+b	+c	+d	=	1
8a	+4b	+2c	+d	=	4

-1a	+b	-1c	+d	=	3
			d	=	2
a	+b	+c	+d	=	1
	-4b	-6c	-7d	=	-4

( -8 times line 3 was added to line 4 )

-1a	+b	-1c	+d	=	3
			d	=	2
	2b		+2d	=	4
	-4b	-6c	-7d	=	-4

( 1 times line 1 was added to line 3 )

a	-1b	+c	-1d	=	-3
			d	=	2
	2b		+2d	=	4
	-4b	-6c	-7d	=	-4

( The 1 line was divided by -1 )

a	-1b	+c	-1d	=	-3
			d	=	2
	2b		+2d	=	4
		-6c	-3d	=	4

( 2 times line 3 was added to line 4 )

a	-1b	+c	-1d	=	-3
	2b		+2d	=	4
			d	=	2
		-6c	-3d	=	4

( the 3 line was interchanged with the 2 line )

a	-1b	+c	-1d	=	-3
	b		+d	=	2
			d	=	2
		-6c	-3d	=	4

( The 2 line was divided by 2 )

a	-1b	+c	-1d	=	-3
	b		+d	=	2
		-6c	-3d	=	4
			d	=	2

( the 4 line was interchanged with the 3 line )

a	-1b	+c	-1d	=	-3
	b		+d	=	2
		c	+0,5d	=	-0,667
			d	=	2

( The 3 line was divided by -6 )

3 line:
Substitute variables already known:
Solve for c :	c = -1,667

2 line:
Substitute variables already known:
Solve for b :	b = 0

1 line:

Substitute variables already known:

-1⋅0

+⋅(-1,667)

-1⋅2

-3

Solve for a :

a = 0,667

Inserting shows that the function equals to ist.

How to find a function with a given inflection point?

An inflection point gives multiple equations: On the one hand, you got the y-value. On the other hand, you know that the second derivative is at an inflection point. Let's take a look at an example for a function of degree having an inflection point at (1|3):

You are looking for a function with:
Function of degree 3
root at 2
root at 4
Inflection point at (1|3)

Mathepower found the following function:

This is the graph of your function.

Dein Browser unterst�tzt den HTML-Canvas-Tag nicht. Hol dir einen neuen. :P

Roots
y-axis intercept at (0|0)
Maximum and minimum turning points
Inflection points

This is how Mathepower calculated:

The point at (1|3) gives the equation :

simplified: :
1a+1b+1c+1d=3

So, we got the following system of equations: :

This is how to solve this system of equations:

1 line:	c+1d = 3
d can be chosen freely
Solve for c : :	c = -1d +3

Inserting shows that the function equals to ist.

And how to use that in my example?

Just enter your exercise above. Mathepower shows how it works by doing a free step-by-step calculation. Or just make up any interesting exercise and check out what Mathepower does.

Can you graph functions on TI 84?

On the TI-83 and TI-84, this is done by going to the function screen by pressing the “Y=” button and entering the function into one of the lines. After the function has been entered, press the “GRAPH” button, and the calculator will draw the graph for you.

How do you calculate functions on a TI 84?

"" = Functions of the form y = f(x) can be entered into the TI-83/TI-84 Plus using the "Y menu. To access the "Y" menu, press the Y key. Type the expres- sion f(x) after Y, using the X,T,0,n key for the variable x and press ENTER.