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

Aug 13, 2015

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

#### Explanation:

To graph $y = {x}^{2}$ and $y = x + 3$

$y - {x}^{2} = 0 \text{ }$ and $\text{ } y - x - 3 = 0$

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 $\left(y - f \left(x\right)\right) g \left(x\right) = 0$ for $y = \frac{f \left(x\right) g \left(x\right)}{g \left(x\right)}$ which is equivalent to $y = f \left(x\right)$ restricted to the domain of $g$ except the zeros of $g$.

So we can restrict the domain of a function $f \left(x\right)$ to and interval $\left(a , b\right)$ by multiplying $\left(y - f \left(x\right)\right) = 0$by a function with domain $\left(a , b\right)$. Like $g \left(x\right) = \sqrt{- \left(x - a\right) \left(x - b\right)}$. (Of course, you can expand the radicand.)

For example, To restrict the domain of $y = {x}^{2}$ to $\left(- 1 , 2\right)$ 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 $\left(a , \infty\right)$ you can use $g \left(x\right) = \sqrt{x - a}$

For example, (y-x^3)sqrt(x+1)=0 restricts the cube to $\left(- 1 , \infty\right)$

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:

$y = 3 x + 2$

$y = - \frac{1}{2} x - 5$

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

$y - 3 x - 2 = 0$

$y + \frac{1}{2} x + 5 = 0$

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

$\left(y - 3 x - 2\right) \left(y + \frac{1}{2} x + 5\right) = 0$

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 $\left(y - f \left(x\right)\right) \left(y - g \left(x\right)\right) = 0$...

#### Explanation:

If you have functions $f \left(x\right)$ and $g \left(x\right)$ then try graphing:

$\left(y - f \left(x\right)\right) \left(y - g \left(x\right)\right) = 0$

That usually works.

For example, $f \left(x\right) = {x}^{2}$, $g \left(x\right) = \sin \left(x\right)$ ...

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...

$y = f \left(x\right)$

$\implies y - f \left(x\right) = 0$

$y = g \left(x\right)$

$y - g \left(x\right) = 0$

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

As this gives you solutions:

$y = f \left(x\right) \mathmr{and} y = g \left(x\right)$

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

This can be obtaine by:

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

$\to$https://www.desmos.com/calculator

This is another method that we can use...

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

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

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 $\left(y + x\right) \left(y - x\right) = {y}^{2} - {x}^{2}$ 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}