Question

What is the equation when y=5x+4 is reflected over the x-axis, y-axis & y=x?

Answer 1

A)
`y=-5x-4`

B)
`y=-5x+4`

C)
`y=(x-4)/(5)`

Explanation:

So we have the equation:

`y=5x+4`

Let's write this in function notation. Thus:

`y=f(x)=5x+4`

A)

To flip a function over the x-axis, multiply the function by -1. Thus:

`f(x)=5x+4\\-(f(x))=-(5x+4)`

Simplify:

`-f(x)=-5x-4`

B) To flip a function over the y-axis, change the variable x to -x. Thus:

`f(x)=5x+4\\f(-x)=5(-x)+4`

Simplify:

`f(-x)=-5x+4`

C) A reflection over the line y=x is synonymous with finding the inverse of the function.

To find the inverse, switch x and f(x) and solve for f(x):

`f(x)=5x+4`

Switch:

`x=5f^(-1)(x)+4`

Subtract 4 from both sides:

`x-4=5f^(-1)(x)`

Divide both sides by 5:

`f^(-1)(x)=(x-4)/(5)`

And we're done :)