Question

What are the x-values on the graph of `y= 1/x` where the graph is parallel to the line `y= -4/9x+7`?

What are the x-values on the graph of `y= 1/x` where the graph is parallel to the line `y= -4/9x+7`?

Answer 1

`x in {-3/2, 3/2}`

Explanation:

This question is actually asking where the tangent lines of `y=1/x` (which can be thought of as the slope at the point of tangency) is parallel to `y=-4/9x+7`. As two lines are parallel when they have the same slope, this is equivalent to asking where `y=1/x` has tangent lines with a slope of `-4/9`.

The slope of the line tangent to `y=f(x)` at `(x_0, f(x_0))` is given by `f'(x_0)`. Together with the above, this means our goal is to solve the equation

`f'(x) = -4/9` where `f(x) = 1/x`.

Taking the derivative, we have

`f'(x) = d/dx1/x = -1/x^2`

Solving,

`-1/x^2 = -4/9`

`=> x^2 = 9/4`

`:. x = +-3/2`