Question

How can I use a graph to write an algebraic function?

Answer 1

From the graph you should be able to determine some `(x,y)` coordinates. Use these point to solve the general equation form for the constants.

Explanation:

The number of point you need will depend upon the shape of the graph: 2 sets for a linear graph; 3 sets for a parabola and so on.

Examples:

1. Linear Graph
graph{(y-(5x-3))(x^2+(y+3)^2-0.01)((x-1)^2+(y-2)^2-0.01)=0 [-5.15, 7.337, -4.03, 2.213]}
2 possible points that could be read from this graph:
`color(white)("XXX")(x,y) in { (0,-3), (1,2)}`

General form of a linear equation (one version):
`color(white)("XXX")y=ax+b`

Using the points we read from the graph for `(x,y)`
`color(white)("XXX")-3=axx(0)+b`
`color(white)("XXX")2=axx(1)+b`
Two equations in two variables can be easily solved as :
`color(white)("XXX")b=-3 and a=5`
So the equation can be written as
`color(white)("XXX")y=5x-3`

2. Parabolic Graph
graph{(y-(3*x^2 -2x +4))((x-1)^2+(y-5)^2-0.01)((x+1)^2+(y-9)^2-0.01)(x^2+(y-4)^2-0.01)=0 [-4.23, 6.87, 3.568, 9.115]}
3 possible points that could be read from this graph:
`color(white)("XXX")(x,y) in {(-1,9), (0,4), (1,5)}`

Using the general form for a parabola of:
`color(white)("XXX")y=ax^2+bx+c`

and the points we read from the graph:
`color(white)("XXX")9=a(-1)^2+b(-1)+c`
`color(white)("XXX")4=a(0)^2+b(0)+c`
`color(white)("XXX")5=a(1)^2+b(1)+c`
Again (without the details) we have 3 equations in 3 variables that can be solved as
`color(white)("XXX")a=3, b=(-2), c=4`
So the equation can be written as
`color(white)("XXX")y=3x^2-2x+4`

More complex polynomials will require more points but the process is identical.