How do you graph two or more functions on the same graph with the graphing utility on Socratic.org?

5 Answers
Aug 13, 2015

Write each equation as an expression = 0. Then set the product of the expressions equal to 00

Explanation:

To graph y=x^2y=x2 and y = x+3y=x+3

y-x^2 = 0" "yx2=0 and " "y-x-3=0 yx3=0

Graph: (y-x^2)(y-x-3)=0

graph{(y-x^2)(y-x-3)=0 [-7.17, 15.33, -2.43, 8.82]}

And if you're patient enough to type this: (you can do one copy and paste) See Edit Below

(y-x^2)(y-x-3)(sqrt(13/4-(x-1/2)^2))/(sqrt(13/4-(x-1/2)^2) ) <= 0

then you can get just the region bounded by the two:

graph{(y-x^2)(y-x-3)(sqrt(13/4-(x-1/2)^2))/(sqrt(13/4-(x-1/2)^2) ) <= 0 [-4.624, 7.864, -0.51, 5.72]}

Edit

It looks like the grapher works by solving (y-f(x))g(x)=0(yf(x))g(x)=0 for y = (f(x)g(x))/(g(x))y=f(x)g(x)g(x) which is equivalent to y = f(x)y=f(x) restricted to the domain of gg except the zeros of gg.

So we can restrict the domain of a function f(x)f(x) to and interval (a,b)(a,b) by multiplying (y-f(x)) = 0(yf(x))=0by a function with domain (a,b)(a,b). Like g(x) = sqrt(-(x-a)(x-b))g(x)=(xa)(xb). (Of course, you can expand the radicand.)

For example, To restrict the domain of y=x^2y=x2 to (-1,2)(1,2) use

(y-x^2)sqrt(-(x+1)(x-2))=0

graph{(y-x^2)sqrt(-(x+1)(x-2))=0 [-5.404, 8.645, -0.9, 6.11]}

To restrict to (a,oo)(a,) you can use g(x) = sqrt(x-a)g(x)=xa

For example, (y-x^3)sqrt(x+1)=0 restricts the cube to (-1,oo)(1,)

graph{(y-x^3)sqrt(x+1)=0 [-7.33, 10.45, -2.37, 6.5]}

Jan 9, 2016

If you wanted to graph the lines:

y=3x+2y=3x+2

y=-1/2x-5y=12x5

The best thing I've seen to do is to manipulate them both so that they're both equal to 0:

y-3x-2=0

y+1/2x+5=0

And then put the equations into the graphing tool as a product of the two equations which equals 0:

(y-3x-2)(y+1/2x+5)=0

Without hashtags:

(y-3x-2)(y+1/2x+5)=0

graph{(y-3x-2)(y+1/2x+5)=0 [-15.55, 12.93, -8.66, 5.58]}

This can be done with more than lines, too:

graph{((x-500)^2+(y-500)^2-500^2)((x-250)^2+(y-750)^2-100^2)((x-750)^2+(y-750)^2-100^2)((y-500)^2/150^2+(x-500)^2/50^2-1)((x-500)^2/200^2+(y-200)^2/75^2-1)=0 [-580, 1644, -100, 1076]}

What went into the grapher:

((x-500)^2+(y-500)^2-500^2)((x-250)^2+(y-750)^2-100^2)((x-750)^2+(y-750)^2-100^2)((y-500)^2/150^2+(x-500)^2/50^2-1)((x-500)^2/200^2+(y-200)^2/75^2-1)=0

Mar 7, 2016

Express as (y-f(x))(y - g(x)) = 0...

Explanation:

If you have functions f(x) and g(x) then try graphing:

(y-f(x))(y-g(x)) = 0

That usually works.

For example, f(x) = x^2, g(x) = sin(x) ...

graph{(y-x^2)(y-sin x) = 0 [-10, 10, -5, 5]}

Dec 29, 2017

As seen below...

Explanation:

There are few ways of doing this, but one way is by using this idea...

Defining your first function...

y = f(x)

=> y - f(x) = 0

Your second function...

y = g(x)

y - g(x) = 0

=> color(red)((y- f(x) ) ( y - g(x) ) = 0

As this gives you solutions:

y = f(x) and y = g(x)

graph{(y - e^x )(y + x^2) = 0 [-5.018, 4.98, -2.04, 2.96]}

This can be obtaine by: enter image source here

Another method is via using 'desmos' a graphing software...

-> https://www.desmos.com/calculator

enter image source here

This is another method that we can use...

This is a good website for also plotting coodinates, if needed...

enter image source here

This is also a great website for solving and plotting inequalities...

enter image source here

Dec 29, 2017

See below

Explanation:

Notice that these 3 commands generate the same graph
graph{x^2 [-10, 10, -5, 5]} graph#{y=x^2 [-10, 10, -5, 5]} graph#{y-x^2=0 [-10, 10, -5, 5]}
If you don't set the display range, it will be set to default
[-10, 10, -5, 5]

Now the magic starts: we multiply 2 expressions that are equal to 0.

For example a parabola and a circle:
graph{(y-x^2)(x^2+y^2-1)=0}
graph{(y-x^2)(x^2+y^2-1)=0}

You can shift and strech at will:
graph{(#color(red)("y-2")-(color(blue)("x+3"))^2)((color(blue)("(x-4)/2"))^2+(color(red)("y+1"))#^2-1)=0}
graph{(y-2-(x+3)^2)(((x-4)/2)^2+(y+1)^2-1)=0}

Simplifying these expressions mathematically doesn't affect graph.

For example if we want to draw lines y+x=0 and y-x=0 we could use (y+x)(y-x)=y^2-x^2 and graph:

graph{y^2-x^2=0}
graph{y^2-x^2=0}

Also we can restrict te domain to basically any subset of XY plane we can imagine. For example a circle with radius 3.

graph{(y^2-x^2)(y^2-4x^2)(4y^2-x^2)sqrt(9-x^2-y^2)=0}
graph{(y^2-x^2)(y^2-4x^2)(4y^2-x^2)sqrt(9-x^2-y^2)=0}