Question

How are the graphs `f(x)=sqrtx` and `g(x)=-sqrt(0.4x)+3` related?

Answer 1

See explanation

Explanation:

`g(x)` is an adjusted (transformation) of `f(x)`

Consider these points individually
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("x->0.4x)`

The multiply by `0.4` 'shifts' any point on `sqrt(x)` to the left
by `sqrt(x-0.6x)` in that `1-0.4=0.6`

In other words it changes the horizontal scale of the plot
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)(+3)`

This raises the `sqrt(x)` by 3 so if `y=sqrt(x)` then `y+3=sqrt(x)+3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Negating "sqrt(x)" "->" "-sqrt(x))`

For this condition it would reflect `+sqrt(x)` about the x-axis in that what was a positive `y` value becomes a negative `y` value.

However we have lifted the whole thing by the addition of 3 so it will in fact be reflected (rotated) about a line parallel to the x-axis.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("The line if reflection")`

If we did not have the + 3 it would be the x-axis. However, by adding 3 we have effectively lifted the line of reflection as well as the graphed line by 3.

Thus as this is not a `ul("direct reflection")` of the original image should we call it such? `color(brown)("It is more like a transformation")`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the point of intersection.")`

Equate `f(x)=g(x)` to find `x_("intersection")`

`sqrt(x)=-sqrt(0.4x)+3`

Changed my mind about how to resolve this.

Write as: `" "sqrt(x)=-sqrt(0.4)sqrt(x)+3`

`sqrt(x)(1+sqrt(0.4))=3`

`x=(3/(1+sqrt(0.4)))^2 = 3.377223.....`

`color(red)("I will let you solve the rest" )`

Substitute the value of `x` in `f(x)` to determine `y_("intersection")` and thus the line if reflection.

Notice that the curve near the y axis of the inverted plot (red) is visibly compressed when compared to the other plot (blue). The other point differences are harder to observe but they are there.